quantum origin of the electroweak scale - helsinki

Master’s Thesis

theoretical physics

Quantum Origin of the Electroweak Scale

Hanna Haataja

2016

Advisor: Kimmo Tuominen

Examiners: Kimmo Tuominen

Aleksi Vuorinen

UNIVERSITY OF HELSINKI

DEPARTMENT OF PHYSICS

PL 64 (Gustaf Hallstromin katu 2)

00014 Helsingin yliopisto

Matemaattis-luonnontieteellinen tiedekunta Fysiikan laitos

Hanna Haataja

Sahkoheikon skaalan kvanttimekaaninen alkupera

Teoreettinen fysiikka

Pro gradu -tutkielma 3/2016 65

Coleman-Weinberg mekanismi, kvanttikenttateoria, aarellisen lampotilan kenttateoria

Kumpulan kampuskirjasto

Pro gradu -tutkielmassa esitellaan Coleman-Weinberg mekanismi kahden esimerkki-

laskun kautta. Tutkielmassa lasketaan efektiivinen potentiaali massattomassa skalaari-

teoriassa ja massattomassa skalaarikvanttielektrodynamiikassa. Esimerkkilaskujen

jalkeen esitellaan yksinkertainen malli, jossa skaalainvarianssin rikkova skalaarihiukka-

nen kuuluu niin kutsuttuun piilotettuun sektoriin.

Ennen esimerkkilaskuja esitellaan kvanttikenttateorian perusteita. Naiden yhteyessa es-

itellaan vuorovaikuttavien kenttien kasittelyyn liittyvat Feynmanin saannot ja

Feynmanin diagrammit.

Lisaksi tutkielmassa esitellaan termisen kenttateorian perusteita ja lasketaan

efektiivinen potentiaali kahdessa tapauksessa: massattomassa skalaariteoriassa ja

Standardimallissa. Jalkimmaisessa tapauksessa fermionien kontribuutio jatetaan

huomioimatta. Esityksessa keskitytaan erityisesti korjaukseen, joka voidaan laskea ns.

rengasdiagrammien avulla. Laskujen motivaationa on vakuumissa

spontaanisti rikkoutuneiden symmetrioiden mahdollinen palautumisesta riittavan

korkeassa lampotilassa ja tasta seuraavat faasimuutokset. Sovelluskohteita ovat

esimerkiksi neutronitahdet ja varhainen maailmankaikkeus.

Tutkielmassa tarkasteltujen menetelmien avulla voidaan analysoida piilotettuja

sektoreita sisaltavia Standardimallin laajennuksia.

Tiedekunta Osasto — Fakultet Sektion — Faculty Section Laitos — Institution — Department

Tekija — Forfattare — Author

Tyon nimi — Arbetets titel — Title

Oppiaine — Laroamne — Subject

Tyon laji — Arbetets art — Level Aika — Datum — Month and year Sivumaara — Sidoantal — Number of pages

Tiivistelma — Referat — Abstract

Avainsanat — Nyckelord — Keywords

Sailytyspaikka — Forvaringsstalle — Where deposited

Muita tietoja — ovriga uppgifter — Additional information

HELSINGIN YLIOPISTO — HELSINGFORS UNIVERSITET — UNIVERSITY OF HELSINKI

Faculty of Science Department of Physics

Hanna Haataja

Quantum origin of the electroweak scale

Theoretical physics

Master’s thesis 3/2016 65

Coleman-Weinberg mechanism, quantum field theory, thermal field theory

Kumpula campus library

In this thesis we introduce the Coleman-Weinberg mechanism through sample calcula-

tions. We calculate the effective potential in the massless scalar theory and massless

quantum electrodynamics. After sample calculations, we walk through simple model in

which the scalar particle, that breaks the scale invariance, resides at the hidden sector.

Before we go into calculations we introduce basic concepts of the quantum field theory.

In that context we discuss interaction of the fields and the Feynman rules for the

Feynman diagrams.

Afterwards we introduce the thermal field theory and calculate the effective potential in

two cases, massive scalar theory and the Standard Model without fermions. We intro-

duce the procedure how to calculate the effective potential, which contains ring diagram

contributions. Motivation for this is knowledge of that sometimes the spontaneously

broken symmetries are restored in the high temperature regime. If the phase transi-

tion between broken-symmetry and full-symmetry phase is first order phase transition

baryogenesis can happen.

Using the methods introduced in this thesis the Standard Model extensions that contain

hidden sectors can be analyzed.

Tiedekunta Osasto — Fakultet Sektion — Faculty Section Laitos — Institution — Department

Tekija — Forfattare — Author

Tyon nimi — Arbetets titel — Title

Oppiaine — Laroamne — Subject

Tyon laji — Arbetets art — Level Aika — Datum — Month and year Sivumaara — Sidoantal — Number of pages

Tiivistelma — Referat — Abstract

Avainsanat — Nyckelord — Keywords

Sailytyspaikka — Forvaringsstalle — Where deposited

Muita tietoja — ovriga uppgifter — Additional information

HELSINGIN YLIOPISTO — HELSINGFORS UNIVERSITET — UNIVERSITY OF HELSINKI

Contents

1 Introduction 2

1.1 Some Notations and Conventions . . . . . . . . . . . . . . . . . . . 2

1.2 Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Basics of Quantum Field Theory 8

2.1 Canonical Quantization of Free Fields . . . . . . . . . . . . . . . . 8

2.2 Perturbation Theory and Interacting Fields . . . . . . . . . . . . . . 14

3 Coleman-Weinberg Mechanism 20

3.1 Massless Scalar Theory . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Massless Scalar Electrodynamics . . . . . . . . . . . . . . . . . . . . 25

3.3 Coleman-Weinberg Mechanism with a Higgs Portal . . . . . . . . . 29

4 Thermal Quantum Field Theory 39

4.1 Reminder of Statistical Physics . . . . . . . . . . . . . . . . . . . . 39

4.2 Path Integral for the Partition Function . . . . . . . . . . . . . . . 39

4.3 Calculation of Polarization Tensor . . . . . . . . . . . . . . . . . . . 41

5 Effective Potential at Finite Temperature 42

5.1 The Scalar Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.2 The Standard Model Without Fermions . . . . . . . . . . . . . . . . 48

5.3 Scalar Theory with the Gauge Field . . . . . . . . . . . . . . . . . . 55

6 Summary and outlook 60

1

Chapter 1

Introduction

1.1 Some Notations and Conventions

Here we introduce some of the most important notations. Although many of

parameters and notations are intended to be introduced as they are encountered.

We work in the units where ~ = c = 1.

When using 4-vectors, x = (x0, ~x), where ~x is normal 3-vector. Using index

notation xµ = (x0, xi), Latin letters as index refer usually to spatial coordinates,

i ∈ 1, 2, 3, and Greek letters refer to space-time, µ ∈ 0, 1, 2, 3. Note that when

refering to specific feature of the field we may use Greeck letters, for example mass

of a scalar field mφ.

The Einstein summation convention will be used, so that, for example, the dot

product of a four-vector k with itself is written as

k · k = ηµνkµkν = (k0)2 − (~k)2,

where ηµν is component of the Minkowski metric.

The Minkowski metric is denoted by η,

η =

1 0 0 0

0 −1 0 0

0 0 −1 0

0 0 0 −1

.

Indecies can be lowered or rised if needed via,

xµ = ηµνxν , xµ = ηµνxν .

General metric is denoted by g and defined if needed.

Time derivative is denoted by shorthand

dx

dt= x

2

1.2. PROLOGUE 3

and all the other derivatives are either shown explicitly or by using shorthand

notation∂

∂xµ= ∂µ and

∂

∂xµ= ∂µ.

In particular

∂µ∂µ = ∂2t −∇2.

1.2 Prologue

The Standard Model of particle physics is a theory that describes elementary par-

ticles and interactions, forces, between them. The theory consists of three forces:

electromagnetism, weak force and strong interactions. Quantum Electrodynam-

ics, QED, is quantum field theory that describes particles that have elecromagnetic

charge. QED introduces photons as force carriers. Quantum Chromodynamics,

QCD, describe the strong interactions, namely interactions between quarks, which

make up protons and neutrons in nucleus. Weak force is responsible for nuclear

β−decay of certain radioactive isotopes. Weak force has three vector bosons as

force carriers W−,W+ and Z, from which first two has electric charge. In the

Stardard Model electromagnetism and weak force has been unified in one field

theory called electroweak theory [1].

Because the Standard Model is a quatized field theory, it does not include

gravitation. Gravitation does not yet have consistent perturbative quantum field

theory form and as such it does not fit in the frame of the Standard Model. Force

carrier of gravitation is called graviton.

So far the Standard Model seems to fit very well to data gathered from exper-

iments. Still there exist phenomena that are not included in the Standard Model.

The indirect observations of the dark matter [2] has been made since 1930s [3],

particles that form the dark matter are yet to be found. Our Universe is expanding

at an increasing rate [4, 5], the source of that expansion is called dark energy [6].

The nature of dark energy is still mostly unknown.

One open question is the latest found particle the Higgs boson [7,8]. The Higgs

boson is essential component of the Standard Model. The found Higgs boson has

a mass of approximately 125 GeV. Still many of the Higgs bonsons properties are

unknown. In the Standard Model the Higgs sector is

LHiggs = (DµH)†(DµH) +VH = (DµH)†(DµH) +µ2SM(H†H)−λH(H†H)2, (1.1)

where µSM is the Higgs mass parameter, λH is the self-coupling constant and H

is complex scalar field. The covariant derivative is

Dµ = ∂µ +ı

2g′Bµ +

ıσa

2gAaµ,

4 CHAPTER 1. INTRODUCTION

where σa are Pauli matrices and index a runs from 1 to 3. We define AµA =

(Aµa , Bµ). The field descibe above are eigenstates of isospin and hypercharge and

thus Lagrangian is symmetric under SU(2)×U(1)Y transformation.

The Higgs mass parameter is rather problematic in the Standard Model. If the

Higgs mass parameter is positive, the potential VH in equation 1.1 has the shape of

the parabola and the vacuum is unique and symmetric under gauge trasformation.

However in the Standard Model the Higgs mass parameter is set to be negative.

The complex Higgs doublet can be expressed with four real fields. So that

H =1√2

(φ3 + ıφ4

φ1 + ıφ2

).

From which the lower component is electromagnetically neutral, but upper compo-

nent has an electromagnetic charge +1. These charge assignments give an eigen-

state for hypercharge and allow the selection of the vacuum state that has U(1)

electromagnetic symmetry by shifting the field φ1. From that shift of the field,

φ1 = v + χ,

the Higgs fields acquire masses,

m21(v) = 3λv2 − µ2

SM ,

m22(v) = m2

3(v) = m24(v) = λv2 − µ2

SM .

The symmetry of the Lagrangian is broken [9]

SU(2)× U(1)Y → U(1)EM .

At the classical minimum 〈v〉0 = µSMλ1/2 the gauge bosons aquire masses. The

gauge boson mass term is of the form AA†µ M2AB(v)AµB, where A and B run from 1

to 4. The mass matrix M2AB is non diagonal

M2(v) =

g2

4v2 0 0 0

0 g2

4v2 0 0

0 0 g2

4v2 −gg′

4v2

0 0 −gg′

4v2 g′2

4v2

. (1.2)

The new fields W+µ , W−

µ , Zµ, and Aµ are defined as a linear combinations of the

field Aaµ and Bµ so that new fields have masses

m2W±(v) = g2

4v2, m2

Z(v) = g2+g′2

4v2 and m2

A(v) = 0.

This mechanism is called the Higgs mechanism [9, 10] through which the gauge

bosons W± and Z aquire masses.

1.2. PROLOGUE 5

The Higgs mass parameter has dimensions of mass and thus it must be directly

proportional to the mass of the Higgs boson. Unlike in the other sections of the

Standard Model the Higgs sector posses no addional symmetry that would guard

the Higgs mass parameter from the quantum corrections. Thus the Higgs mass

receives quadratic corrections [11]

m2H → m2

H +3Λ2

16π2v2(m2

H +m2Z + 2m2

W − 4m2t ), (1.3)

where m2Z is the mass of the Z -boson, m2

W is the mass of the W -boson and m2t is

the mass of the top quark.The cutoff parameter Λ should be as high as it can within

the validity of the theory. The Standard Model is valid as long as gravitational

effects are neglectable, therefore Λ is same magnitude as Planck mass Mpl ∼ 1019

GeV. Thus all the dependence in the cutoff in the equation 1.3 must cancel out.

By expressing masses in equation 1.3 in means of coupling constants and vacuum

expectation value of the Higgs field we obtain the relation [12] among the coupling

constants3

2g2

1 +9

2g2

2 + 6λH = 4∑f

h2f , (1.4)

where g21 and g2

2 are the U(1) and SU(2) gauge couplings and hf are the Yukawa

couplings. This means some of the Standard Model parameters must be fine tuned

to achieve desired result.

In the Standard Model this separations of the scales, the electroweak scale and

the gravity, is achieved through fine tuning. Naturalness [13] is principle that

nature does not have conspiracies between phenomena occurring at very different

length scales [14].Theories that contain scalar fields present unnaturalness because

the mass of the scalar field highly depends in the cutoff of the theory [15]. Other

perhaps more elegant solutions, than fine tuning, to the hierarchy problem, why

weak force is so much stronger gravity, have been proposed for long now.

Perhaps most well known solution of the hierarchy problem are theories that

include supersymmetry. In these supersymmetric theories every Standard Model

particle has supersymmetric partner which differs from the Standard Model par-

ticle only by half-integer spin. Meaning that every fermion has boson as a super-

partner and every boson has fermionic superpartner. The quantum corrections of

the superpartners cancels with the contribution from the Standard Model sector in

the calculation of the radiative corrections to the Higgs boson mass [16]. The hier-

archy between the electroweak scale and the Planck scale is achieved in a natural

manner.

It has been proposed that the Higgs boson is not actually elementary particle,

but a bound state of new strong force [17]. The composite Higgs can be responsible

for electroweak symmetry breaking and new particles are expected to have mass

around TeV scale. In composite Higgs theories an approximate global symmetry

protects the Higgs mass [18].

6 CHAPTER 1. INTRODUCTION

Downfall of the mentioned solutions is the way in which they introduce new

particles. Aside from the Higgs boson no new particles have been found. Even

though these theories suggest that new particles should be found at energies avail-

able at the Large Hadron Collider [19].

There has been lot of interest in theories where the Higgs boson would be

pseudo-Nambu-Goldstone boson. In these so called little Higgs theories the origin

of the Higgs mass would be much like pions, that are preudo-Nambu-Goldstone

bosons. Pion is the result of spontaneous and explicit breakdown of the chiral

flavor symmetry, so that pion is not massless as Nambu-Goldstone boson would

be [20]. Same way for the Higgs, a global symmetry is broken spontaneously and

then broken explicitly when two or more couplings in the Lagrangian are non-

vanishing, thus generating light but massive Higgs boson [21]. The idea was first

suggested in 1970s [22] but the first successful little Higgs model was publiched in

2001 [23] and after that lots of little Higgs models have been studied [21].

The idea that the Higgs particle may originate from extra-dimensional com-

ponents of gauge fields originates from the late 70s [24]. In these theories the

electroweak symmetry itself protects the Higgs potential from divergences [25].

The exitence of additional dimensions are highly districted and some indicators

should be found in LHC [26].

In this thesis we consider scale independent theories, which means µSM ≡ 0 at

the tree level. When the Higgs mass parameter is set to zero the Standard Model

Lagrangian has no dimensionful parameters [19]. The idea to use scale invariance

to protect the Higgs mass was first discussed in 1995 [27] and after discovey of

the Higgs boson it has gained popularity. As we set the Higgs mass parameter to

zero, we lose source of the symmetry break. Hoewever the symmetry break can be

obtained dynamically by Coleman-Weinberg mechanism [28]. Coleman-Weinberg

mechanism applied to the Standard Model alone lead to a Higgs boson mass that

is smaller than the gauge bosons, thus leading to theory that cannot explain ob-

served values. Coleman-Weinberg mechanism however opens up extensions to the

Standard Model. In the extensions idea is that the Higgs field is coupled to scalar

in so called hidden sector. The symmetry is broken in the hidden sector and then

conveyed to the scale invariant Standard Model via interaction with the Higgs

sector. Hidden sectors arise in many top-down models, including those inspired by

the brane world scenario and string theory. Hidden sector is relevant as a source

of dark matter.

To have a Universe where in nature consist of particles, means that in the

early Universe particle-antiparticle symmetry must be violated. The process that

produces this antisymmetry between baryons and antibaryons is called baryogene-

sis [29]. All theories attempting to offer an explanation for baryogenesis have to ful-

fill so called Sakharov’s conditions [30]. One of the Sakharov’s conditions requires

that the electroweak symmetry breaking must be a first order phase transition [31].

In Standard Model the electroweak phase transition is a smooth crossover. Addi-

1.3. STRUCTURE OF THE THESIS 7

tional degrees of freedom can lead to a strongly first order transition.

It has been observed that symmetry can be recovered in the thermal field the-

ory. At some high enough temperature there may occur phase transition that

takes system from the broken-symmetry phase to the full-symmetry phase. To

study these phase transitions we introduce the prosedure calculating loop correc-

tions in the thermal field theory. The main goal is to study the order of the

phase transition: in many extensions of the Standard Model a strongly first order

transition can be obtained.

1.3 Structure of the Thesis

In chapter 2, we review the basics of the quantum field theory, so that later dis-

cussion would be easy to follow. We start by quantatization of the scalar field

and the electromagnetic field. Followed by introduction of the interactions, in the

scalar theory and the scalar quantum electrodynamics. We introduce the Feynman

diagrams and the loop expansion. In the end of the chapter 2, we briefly discuss

renormalization and effective theories.

In chapter 3, we study in detail how Coleman-Weinberg mechanism works, in

two different but very similar cases. After that we introduce the idea how Coleman-

Weinberg machanism would solve the naturalness issue by reviewing a simple

model, in which symmetry break happens in dark sector through the Coleman-

Weinberg mechanism and is conveyed to the Standard Model side through so

called Higgs portal.

In chapter 4, we review the thermal field theory very briefly. We intorduce the

path integral formalism and show how to get the partition function for the free

scalar fields.

In chapter 5, we consider applications for the thermal field theory. We calculate

the effective potential for two cases, first for the massive scalar theory and then

for the Standard model without fermions. Both of these cases has some initial

symmetry that is broken spontatenously at the zero temperature. When consider-

ing the thermal field theory we can see that this symmetry is now regained when

temperature is raised high enough. When this happens there is first-order phase

transition between the broken symmetry-phase and the full-symmetry phase.

In chapter 6 we summarize the thesis and present a brief outlook of possible

directions for further exploration.

Chapter 2

Basics of quantum field theory

This chapter is reminder of the quantum field theory. Many books has been written

about this subject, such as [32] and [33]. Reader should at least be familiar with

classical fields, Lagrangian and Hamiltonian formalism for the fields and quantum

mechanics beforehand.

2.1 Canonical Quantization of Free Fields

In this section we breafly discuss quantatization of free scalar and electromagnetic

fields. We will show how to derive propagators for those two fields.

2.1.1 Canonical Quantatization of Scalar Fields

In quantum mechanics, canonical quantization is a recipe that takes us from the

Hamiltonian formalism of classical dynamics to the quantum theory. The recipe

tells us to take the generalized coordinates qa and their conjugate momenta pa and

promote them to operators. The Poisson bracket structure of classical mechanics

morphs into the structure of commutation relations between operators, so that,

[qi, pj] = ıδij;

[qi, qj] = [pi, pj] = 0.

In field theory we now do the same [33]. Thus a quantum field is and operator

valued function of space obeying the commutation relations

[φa(~x), πb(~y)] = ıδ(3)(~x− ~y)δba;

[φa(~x), φb(~y)] = [πa(~x), πb(~y)] = 0.(2.1)

We have now worked in Schrodinger picture so that our operators, φa(~x) and

πa(~x), do not depend on time. Time dependence is in states |ψ〉 which evolve by

the usual Schrodinger equation

ıd|φ〉dt

= H|φ〉.

8

2.1. CANONICAL QUANTIZATION OF FREE FIELDS 9

In this notation we must be careful, because the notation is deceptive. If we were

to write down the wavefunction in quantum field theory, it would be function of

every possible configuration of the field φ.

Hamiltonian is now operator. We want to now know what is the spectrum of

Hamiltonian. Let’s consider free real scalar field φ(~x, t) whose Lagrangian density

is

L =1

2∂µφ∂

µφ− 1

2m2φ2,

by Euler-Lagrange equations of motions we get Klein-Gordon equation,

∂µ∂µφ+m2φ = 0. (2.2)

Now we can expand the field in Fourier space

φ(~x, t) =

∫d3p

(2π)3eı~p·~xφ(~p, t).

Putting this back to Klein-Gordon equation 2.2 means that φ(~p, t) must satisfy(∂2t + (~p2 +m2)

)φ(~p, t) = 0.

Solutions to this differential equation are harmonic oscillations. That means that

for each value of ~p, φ(~p, t) solves the equation of a harmonic oscillator vibration

at frequency

ω~p =√~p2 +m2.

To quantize φ(~x, t) we must simply quantize infinite nuber of harmonic oscillators

[33]. This can be done by using the creation and annihilation operators, in the same

manner as in the quantum mechanics. Thus we find that spectrum of Hamiltonian

is (n+ 12)ω and fields can be treaded as independent oscillations

φ(~x) =

∫d3p

(2π)3

1√2ω~p

(a~p + a†−~p)eı~p·~x,

π(~x) = −ı∫

d3p

(2π)3

√ω~p2

(a~p − a†−~p)eı~p·~x,

with commutation relation

[a~p, a†~p′ ] = (2π)3δ(3)(~p− ~p′). (2.3)

Finally we get the Hamiltonian in terms of ladder operators

H = 12

∫d3x(π2 + (∇φ)2 +m2φ2)

=∫

d3p(2π)3

ω~p

(a†~p a~p + 1

2[a~p, a

†~p]).

(2.4)

The second term of the Hamiltonian is an infinite c-nuber. It is the sum over all

modes of the zero-point energies, so its understandable why this infinity occurs in

10 CHAPTER 2. BASICS OF QUANTUM FIELD THEORY

our calculation. Fortunately this infinite energy shift is not meaningful in physics,

since experiments measure only energy differences from the ground state of H.

Therefore the second term can be dropped without any other consernes [32].

Operator a†~p creates momentum ~p and energy ω~p. These exitations the creation

opearator creates are called particles. Particles are discrete entities that have

proper relativistic energy-moemntum relation, thats why ω can from now on called

energy, E~p1.

2.1.2 Propagator and Causality

We now move to Heisenberg picture and derive the propagator for scalar field

keeping in mind that our goal is to introduce the Feynman rules.

In Heisenberg picture the operators φ and π are time-dependent as in usual

quantum mechanics

φ(x) = φ(~x, t) = eıHtφ(~x)e−ıHt,

and in similar manner for the π(x) = π(~x, t). The Heisenberg equation of motion

for general operator O,

ı∂

∂tO = [O,H], (2.5)

Heisenberg equation of motion allows us to compute the time depence of φ andπ

explicitly:ı ∂∂tφ(~x, t) = ıπ(~x, t),

ı ∂∂tπ(~x, t) = −ı(−∇2 +m2)φ(~x, t).

Combining these only gives us the Klein-Gordon equation 2.2. It is sensible that

we get same equations of motions in both pictures.

Better understanding of the time dependence we get by expanding φ(x) and

π(x) in terms of creation and annihilation operators. Before we do that we must

note that

Hap = ap(H − Ep),

this can be easily shown by opearating to the one particle state with the operator.

From this follows that

Hnap = ap(H − Ep)n,

for any n. Similar result can be derived for creation operator:

Hna†p = a†p(H + Ep)n.

By using these equatons above we can now derive the time dependent ladder

operators,

eıHtape−ıHt = ape

−ıEpt and eıHta†pe−ıHt = a†pe

ıEpt. (2.6)

1Or just shortly E.


Now we can rewrite the fields in the time dependent form

φ(~x) =

∫d3p

(2π)3

1√2E~p

(a~pe−ı p·x + a†−~pe

ı p·x), (2.7)

π(~x, t) =∂

∂tφ(~x, t),

where p · x is product of two four-vectors.

Now we move to the question of causality. In the Heisenberg picture the am-

plitude for a particle to propagate from x to y is 〈0|φ(x)φ(y)|0〉, where |0〉 is the

vacuum state. Let’s call this creature D(x− y). Each of the operator φ is a sum

of a and a†, but only the term 〈0|a~p a†~q|0〉 = (2π)3δ(3)(~p− ~q) survives. So that the

amplitude gets the form

〈0|φ(x)φ(y)|0〉 = D(x− y) =

∫d3p

(2π)3

1

2Ee−ıp·(x−y). (2.8)

This form of the integral is Lorentz invariant [32]. Now we study this integral for

some particular values of x− y.

Let us consider case where x− y is purely spatial: x0 − y0 = 0 and ~x− ~y = ~r.

The amplitude is then

D(x− y) = 1(2π)3

∫∞0d3p 1

2Eeı~p·~r

= −ı2(2π)2r

∫∞−∞ dp

p√p2+m2

eı pr

= −ı4π2r

∫∞mdρ ρ√

ρ2−m2e−ρr ∼ e−mr,

where ρ = −ıp. Problem in this solution is that even though the propagation

amplitude is exponentially vanishing it is not zero. That means formally that

particle can propagate over spacelike intervals.

However we should not ask if particle can propagate over spacelike intervals,

but whether a measurement performed at one point can affect a measurement at

another point whose separation from the first point is spacelike. Easiest thing to

measure is the field itself. If commutator, [φ(x), φ(y)], vanishes, one measurement

cannot affect the other. We already know that for spacelike interval, x0−y0 = 0 the

commutator goes to zero, because of canonical commutation relations 2.1. We can

look it little closer and more generally determine the character of this commutator

[φ(x), φ(y)] = 1(2π)6

∫d3p 1

Ep

∫d3q 1

Eq

[(a~pe

−ıp·x + a†~peıp·x), (a~qe

−ıq·x + a†~qeıq·x)

]=∫

d3p(2π)3

12Ep

(e−ıp·(x−y) − eıp·(x−y)

)= D(x− y)−D(y − x). (2.9)


When we are talking of the spacelike interval (x−y)2 < 0, we can perform a Lorentz

transformation on the second therm of 2.9, taking (x − y) → −(x − y). The two

terms are now equal and commutator gives zero. That means that causality is

preserved [32].

Convenient way to derive expression for the Feynman propagator is define

quantity

DR(x− y) ≡ θ(x0 − y0)〈0|[φ(x)φ(y)]|0〉,

where θ(x − y) is the Heaviside step function. This entity is really a retarded

Green’s function of the Klein-Gordon operator since

(∂2 +m2)DR(x− y) = −ıδ(4)(x− y).

To get expression for the DR we express it as Fourier integral

DR(x− y) =

∫d4p

(2π)4DR(p)e−ip(x−y),

now the Klein-Gordon equation tells us that

DR(p) =i

p2 −m2.

So for the retarded Green’s function we get

DR(x− y) =

∫d4p

(2π)4

i

p2 −m2 + ıεe−ip(x−y). (2.10)

Since the p0 integral has to be evaluated by contour integral we need to add term

iε [32], in many cases we can take the limit where ε→ 0. Thus we get

DR(x− y) =

∫d4p

(2π)4

i

p2 −m2e−ip(x−y). (2.11)

DR(x−y) is the Feynman propagator for the scalar particle, from now on denoted

by DF (x− y).


2.1.3 Quantatization of Electromagnetic Field

The quantatization of the electromagnetic field is fairly easy task through func-

tional methods, we will not go deeply in those methods of calculation, but bring

up some difficulties and the results.

First let us look at the action of the free electromagnetic field

S =

∫d4x− 1

4(Fµν)

2,

where Fµν is the field strength tensor

Fµν = ∂µAν − ∂νAµ,

where Aµ is the electromagnetic field. Integrating by parts and and expanding

field as a Fourier integral we get

1

2

∫d4k

(2π)4Aµ(k)(−k2ηµν + kµkν)Aν(−k). (2.12)

This expression however vanishes Aµ(k) = kµα(k), for any α(k). The equation

that would then define the Feynman propagator, DνλF ,

(−k2ηµν + kµkν)DνλF (k) = ıδλµ,

has no solution since matrix (−k2ηµν + kµkν) is singular [32].

This difficulty arises form the gauge invariance. The field strength tensor, Fµν ,

is invariant under a general gauge transormation of the form

Aµ(x)→ Aµ(x) +1

e∂µα(x).

Through the functional methods we gain a new term dependence of the gauge

parameter, ξ [32]. Now the equation that defines the Feynman propagator becomes

(−k2ηµν + (1− 1

ξ)kµkν)Dνλ

F (k) = ıδλµ,

which has solution

DµνF (k) = − ı

k2 + ıε

(ηµν − (1− ξ)k

µkν

k2

).

Calculation are often done with specific choice of the parameter ξ. In this thesis

we work in the Landau gauge and therefore our gauge parameter is zero. Thus we

get the propagator for the electomagnetic field

DµνF (k) = − ı

k2 + ıε

(ηµν − kµkν

k2

), (2.13)

where ηµν is the Minkowski metric.


2.2 Perturbation Theory and Interacting Fields

We will discuss two cases which are relevant for the later discussion. First we

introduce the φ4-theory and in that context we study basic terms. Secondly we

discuss scalar quantum electrodynamics.

So far we have discussed the free fields, there has been no interaction between

the particles. To obtain more realistic description of the world, we must include

new terms to the Hamiltonian, or in the Lagrangian as they play the same role.

These nonlinear terms will couple different Fourier modes to one another. To pre-

serve causality, we insist that the new terms may involve only products of fields at

the same spacetime point [32]. So that [φ(x)]4, is fine, but φ(x)φ(y), is not allowed.

Interaction partof the Hamiltonian is

Hint =

∫d3xHint(φ(x)) = −

∫d3xLint(φ(x)).

For now on we can restrict ourselves to theories in which Lint is a function only of

the fields, not of their derivates.

For φ4-theory Lagrangian density,

L =1

2(∂µφ)2 − 1

2m2φ2 − λ

4!φ4, (2.14)

where λ is dimensionless coupling constant. The equation of motion for this theory

is

(∂2 +m2)φ = − λ3!φ3,

which cannot be solved by Fourier analysis as the free Klein-Gordon equation 2.2

could be. However the field can be quantized by the same canonical relations 2.1,

since

π =∂

∂φL = φ,

is the same as in the free theory.

For the theories that include only scalar, the allowed interactions are µφ3 andλφ4.

The coupling constant µ has dimensions of mass and λ is dimensionless. Terms

of the form φn, where n > 4 are not allowed, since their coupling constants would

have to be dimension 4−n, those kind of interaction terms are not renormalizable.

More complex theories can be buld by adding several scalar fields, real or complex.

Plausible interactions between the fields are limited by axiom that physical

theories should be renormalizable. Higher order term in the pertubation theory,

will involve integrals ovet the 4-momenta of the intermediate particles. These

integrals are often formally divergent and it is normally treated by introducing

some form of cutoff prosedure [32], the simplest thing is just cut the integral from

some large but finite momentum Λ. At the end of the calculation we can take the

limit Λ→∞, and hope that the physical quantities do not depend on Λ. If physical

quantities are independent of Λ, then the theory is said to be renormalizable.

2.2. PERTURBATION THEORY AND INTERACTING FIELDS 15

In our case renormalization goes through the same steps every time. We start

by noting that the coupling constants and masses we encounter in the Lagrangian

are so called bare values, not physical quantities. We add counter terms to our

Lagrangian. We cut our divergent integrals from some large but finite Λ. And

propose definitons to the physical quantities and work out counter terms. In the

end we hope we have managed to get rid of all the Λ dependence on our effective

potential.

Scalar quantum electrodynamics is considered often to be less important. Our

later theories are based on the scalar quantum electrodynamics so we are going to

study it here. Lagrangian density is

L = |Dµφ|2 −m2|φ|2,

where Dµ = ∂µ + ıeAµ(x) is the gauge covariant derivative. This theory contains

terms like eAµφ∂µφ∗ and e2|φ|2A2. Here e is free dimensionless parameter of the

theory.

To study theories we treat the interaction term Lint as a perturbation, compute

its effects as far in perturbation theory as practical, and hope that the coupling

constant is small enough that this gives a reasonable approximation to the exact

answer. This is obtained through the use of Feynman diagrams and from the

stucture of the perturbation theory it will be possible to at least visualize the

effects of interactions.

This simplification of the perturbation series for relativistic field theories was

the great advance of Tomonaga, Schwinger and Feynman. To achieve this simpli-

fication, each, independently, found a way to reformulate quantum machanics to

remove the special role of time, and then applied his new viewpoint to recast each

term of the perturbation expansion as a spacetime process [32].

In here we skip a little further and go straight to the Feynman diagrams,

without longer derivation. Derivation goes through finding perturbative expansion

for two-point correlation functions, and Wick’s theorem. We will not be discussing

these matters in this thesis, reader who is not familiar with these subjects and

wants to know, can study further from introductory texts on quantum field theory,

such as ref [32].


2.2.1 Feynman Diagrams

We now just state conclusions of the Wick’s theorem without proof or derivation,

it can be found in numerous quantum field introduction books. Wick’s theorem

states that any correlation function of the form 〈0|Tφ1φ2φ3φ4|0〉, where φi = φ(xi),

can be turned into a sum of products of Feynman propagators. The correlation

function gets the form

〈0|Tφ1φ2φ3φ4|0〉 = DF (x1 − x2)DF (x3 − x4)

+ DF (x1 − x3)DF (x2 − x4)

+ DF (x1 − x4)DF (x2 − x3)

(2.15)

In the diagrammatic form, every point from x1 to x4 is represented by a dot,

and each DF (x− y) is a line joining x to y. So that means we have sum of three

diagrams [32]

These diagrams are interpred as two particles are created at one of the spacetime

points, each propagates to one of the other points, and then they are annihilated.

This can happen in three ways, corresponding to the three ways to connect the

points in pairs. The total amplitude for the process is the sum of the three dia-

grams.

Diagrams are not exatly a measurable quantity, but from them we can write

down the total amplitude. This closely resembles the superposition principle in

quantum mechanics, which states that if process can happen multiple diffenrent

ways the total amplitude is the sum of the amplitudes of the individual processes.

These diagrams become interesting when there is more than one field at the

same spacetime point. To achieve this we need to add the interactions. Let us add

the φ4-interaction and we obtain terms like

〈0|T[φ(x)φ(y)(−ı)

∫d4z λ

4!φ4]|0〉

= 3−ıλ4!DF (x− y)

∫d4zDF (z − z)DF (z − z)

+12−ıλ4!

∫d4zDF (x− z)DF (y − z)DF (z − z).

(2.16)


x y

(a) (b) (c)

Figure 2.1: The Feynman diagrams.

This corresponds to the diagrams in figure 2.1. Diagrams 2.1a and 2.1b refer to

the first term propagators and 2.1c refers to the second term.

From these diagrams and equation 2.16 we can now deduce the Feynman rules

from the φ4-theory. So that from the diagram one can easily write down the am-

plitude.

1. For every propagator from x to y, add DF (x− y)

2. for each vertex, add −ıλ∫d4z

3. for each external point, multiply by 1.

4. multiply by the symmetry factor of the diagram Sf . For the exple diagrams

the symmety factors are: for the first term Sf = 34!

and the second term Sf = 124!

.

These are the Feynman rules in the position space, but normally we want to

express these on the momentum-space. These are obtained by expressing the prop-

agator in the Fourier integral.

1. For every propagator with momentum p, add DF (p) = ıp2−m2+ıε

2. for each vertex add −ıλ

3. for each external point, multiply by e−ı·x.

4. At each vertex take care of the momentum conservation,

5. integrate over every unknown momentum q,∫

d4q(2π)4

,

6. multiply by the symmetry factor of the diagram Sf .

These are the Feynman rules that we will use later to determine loop contributions

for the effective potential.

Visually the φ4-theory consists of only the vertices show in figure, and all the

propagators are straight lines representing DF (p).

The Feynman rules for scalar electrodynamics can be found in the same manner.

Scalar quantum electrodynamics contains different vertices. Ones shown in the

figure 2.2, that is called the scalar self interaction vertices. When our scalar field

is complex, that can be expressed as two real fields, φ1 and φ2, same kind of vertex

represents interaction between the real fields, in that case half of the propagators


Figure 2.2: The vertex in the φ4-theory.

would be φ1-propagators and the other half φ2-propagators. The difference here

are the vertices that include the photon propagator, DµνF (k) = − ı

k2+ıε

(ηµν− kµkν

k2

)represented by wavy lines, shown in the figure 2.3. The vertex carries the factor

2ıe2ηµν . The vertex shown in the figure 2.3 illustrates the interaction term e2|φ|2A2.

Figure 2.3: Interaction vertex.

Scalar quantum electrodynamics has different vertex corresponding to the in-

teractions between the scalar and the photon, that corresponds to a term of form

eAµφ∂µφ∗, shown in the figure 2.4. That vertex does not play a role in our later

discussion.

Figure 2.4: Second interaction vertex.

2.2.2 The Loop Expansion

The calculation of the effective potential requires summation of infinite amount

of the Feynman diagrams, that is still beyond calculation abilities. Thus we need

sensible approximation for the effective potential. For that we need loop expansion.

First we introduce parameter a into the Lagrange density, by defining

L(φ, ∂µφ, a) = a−1L(φ, ∂µφ). Now propagator carries factor of a and every vertex

carries a−1. Let P be the power of a assosiated with any graph, then P can be

evaluated

P = I − V,where I is the number of the internal lines and V is the number of vertices. Number

of loops is defined by the number of independent intergration momenta. Every


internal line contributes one integration momentum, but every vertex contributes

delta-function that reduces the number of independent momenta by one, exept for

one delta-function that takes care of over-all momentum conservation. Thus we

get that the number of loops in the graph

L = I − V + 1.

Putting these together we get that

P = L− 1.

Thus we can think the loop expansion - where we first sum all the diagrams that

have zero loops, then sum all the diagrams with one loop and then those with

two loops etc. - as a power-series expansion in a. Of course, each stage in this

expansion also involves an infinite summation, but as we later see that sum can

be easily evaluated. The point here is not that a should be small but rather that

it multiplies the whole Lagrange density, thus being unaffected by the shift of the

field.

2.2.3 What are the Effective Theories?

The effective theories are theories that are low energy limit of the fundametal

theory. If we would know the underlying fundamental theory, as a low energy

limit we could derive our effective theory. The fact is that we do not really know

how nature aroud us works in the terms of mathematics, so we need something to

approach that fundamental theory.

It could be that the fundamental theory of the nature is not at all field theory

but something more complex like string theory and our Standard Model of particle

physics is just an effective theory for that [34].

Chapter 3

Coleman-Weinberg Mechanism

In this chapter we introduce Coleman-Weinberg mechanism. In March 1973 Sind-

ney Coleman and Eric Weinberg published article Radiative Corrections as the

Origin of Spontaneous Symmetry Breaking where they introduce the idea that

radiative corrections produces the spontanious symmetry breaking. According to

their findings massless scalar theory is the simplest theory where this happens [28].

As illustrative examples they showed how to calculate effective potentials in mass-

less scalar field theory and in massless scalar quantum electrodynamics. We will

now retrace those steps.

After we have studied the Coleman-Weinberg mechanism in detail, we put it

in use. First we discuss the mechanism in the context of the Standard Model, but

pretty fast we found that Coleman-Weinberg mechanism alone cannot explain the

mass of the Higgs boson.

Lately Coleman-Weinberg mechanism coupled with classical scale invariance

has gained more popularity. A lot of models [27, 35–39] have been constructed

and studied ranging from fairly simple to more complex ones. Main idea here is

to have structure in the hidden sector which interacts with the Standard Model

sector via interactions with the Higgs field. The symmetry is then broken in the

hidden sector and conveyed to the Standard Model generating the electroweak

scale. Many aspects of these theories can be then studied phenomenologically to

determine if the model fits in the observations.

Hidden sectors arise in many top-down models, including those inspired by

the brane world scenario and string theory. And therefore most discussion of

hidden sector associates the hidden particles with a very high mass scale, and their

couplings to the visible Standard Model sector are often through nonrenormalizable

or loop effects. Hidden sectors do not need to be something complex, like string

theory, but just non-visible quantum field theory [40]. With lower masses and

lower enegy scales we could use the hidden sector as a assitance for the theory. It

has been noticed that through renormalizable interactions, the hidden sector can

be probed at energies available at the Large Hadron Collider [37].

We will introduce one of those models [41] in detail. The chosen model is easily

20

3.1. MASSLESS SCALAR THEORY 21

handled and shows really well how models, which include hidden sector, work.

After the model we introduce the phenomenological review of the authors [41].

3.1 Massless Scalar Theory

Massless scalar theory is φ4-theory in which the renormalized mass is considered

to be zero, bare mass cannot be set to be zero, but term which contains m0 is

included in counterterm.

3.1.1 Effective potential

We start by considering vertices that occur in our theory. In massless scalar theory

the Lagrangian density is

L =1

2(∂νφ)2 +

λ

4!φ4 + counterterms =

1

2(∂νφ)2 +

λ

4!φ4 +

1

2A(∂νφ)2− B

4!φ4− C

2φ2.

The factors A, B and C are the usual wave-function, coupling constant and mass

renormalization counterterms. They are determined self-consistently, order by

order in the expansion, by imposing definitions of the scale of the renormalized

field, the renormalized coupling constant and the renormalized mass. The mass

renormalization term must be included even though we are discussing the massless

theory. This is because the theory posses no such symmetry that would guarantee

vanishing bare mass in the limit of vanisihing renormalized mass [28].

At the tree level there is only one graph that contributes to the potential,

shown in figure 3.1. Therefore potential is

V0(φ) =λ

4!φ4.

Figure 3.1: Tree level Feynman diagram for the massless scalar theory.

One loop corrections come from summing up 1PI-diagrams with one loop and zero

external momentum, shown in the fig. 3.2, as well as contibutions from the mass

and the coupling constant counterterms.

22 CHAPTER 3. COLEMAN-WEINBERG MECHANISM

Figure 3.2: one-loop irreducible diagrams of scalar particles [28]

In massless theory the scalar propagator takes the form

DF (k) =1

k2 + ıε,

nth diagram contains n-numbers of internal propagators and vertices. Symmetry

factor contains rotation and reflection of n-sided polygon:

Sn =1

2n,

external legs are treated as identical since they all carry same momentum, which

is zero.

Using these rules we get correction term

V1(φ) = ı

∫d4k

(2π)4

∞∑n=1

1

2n

( λφ2c

2(k2 + ıε)

)n. (3.1)

We can now take the limit where ε goes to zero and the sum can be calculated

∞∑n=1

1

2n

(λφ2c

2k2

)n= −1

2ln(

1− λφ2c

2k2

).

Thus the correction term yields to

⇒ V1 = − ı2

∫d4k

(2π)4ln(

1− λφ2c

2k2

). (3.2)

To calculate this integral we need to do the Wick-rotation from Minkowski

space to euclidean, k2 = k20 −

∑i k

2i . So the momentum changes are k0 = ık0

E and

ki = kiE.1 The correction to the potential now is

⇒ V1 =1

2

∫d4k

(2π)4ln(

1 +λφ2

c

2k2

), (3.3)

1We now drop the subscript E from euclidian vector k.

3.1. MASSLESS SCALAR THEORY 23

⇒ V1 =1

2

∫d4k

(2π)4ln(

1 +λφ2

c

2k2

)=

1

2

1

(2π)4

∫dΩ4

∫dk k3 ln

(1 +

λφ2c

2k2

).

The angular part of integral can be solved with area of the 4-dimensional sphere,

1

(2π)4

∫dΩ4 =

1

23π2Γ(2)=

1

23π2.

Thus the integral becomes

⇒ V1 =1

24π2

∫dk k3 ln

(1 +

λφ2c

2k2

).

Cutting this divergent integral at k2 = Λ2 we obtain first

V1 =1

24π2

[Λ2

2V ′′0 (φc) +

V ′′0 (φc)2

4

(ln(V ′′0 (φc)

Λ2

)− 1

2

)], (3.4)

where we have thrown away every terms that vanish when our cutoff parameter

Λ2 goes to infinity [28]. Now V ′′0 (φc) = λ2φ2c , and therefore loop corrections are of

the form

V1 =λΛ2

64π2φ2c +

λ2φ4c

256π2

[ln(λφ2

c

2Λ2

)− 1

2

]. (3.5)

No we have reached the effective potential. After this we need to consider renor-

malization conditions and solve for the counterterms. The effective potential now

is

V (φ) =λ

4!φ4 − B

4!φ4 − C

2φ2 +

λΛ2

64π2φ2 +

λ2φ4

256π2

[ln(λφ2

2Λ2

)− 1

2

]. (3.6)

We are still talking about massless theory therefore renormalized mass of scalar is

still kept at zero,

m2 :=∂2

∂φ2c

V (φ)|φ=0 = 0. (3.7)

From which follows that

C =λΛ2

32π2.

The effective potential is

⇒ V (φc) =λφ24φ4 − B

4!φ4c +

λ2φ4c

256π2

[ln(λφ2

c

2Λ2

)− 1

2

]. (3.8)

Unfortunately the renormalized coupling constant cannot be determined as

usually; the fourth derivative of V at the origin does not exist, because of the

logarithmic infrafed singularity [28]. Here we solve this problem, as Coleman

and Weinberg did, by defining the coupling constant at a point away from the

singularity in classical-field space. So we define renormalized coupling constant by

λ :=∂4

∂φ4c

V (φc)|φ=M , (3.9)


where M is just a number with the dimensions of a mass. the parameter M is

completely arbitrary, different choices for M will lead to different definitions of the

theory. In this case the renormalized coupling constat is

∂4

∂φ4c

V (φc) = λ− 3B +11λ2

32π2+

λ2

32π2ln(λM2

2Λ2

)= λ. (3.10)

Any nonzero M is as good as any other. This leads the coupling constant

counterterm

B =λ2

32π2

[ln(λM2

2Λ2

)+

11

3

].

Thus we get our potential to be

⇒ V (φc) =λφ4!φ4 − λ2

256π2φ4c

[ln(λM2

2Λ2

)+

11

3

]+

λ2φ4c

256π2

[ln(λφ2

c

2Λ2

)− 1

2

],

⇒ V (φ) =λ

4!φ4 +

λ2φ4

256π2

[ln( φ2

M2

)− 25

6

]. (3.11)

This is now the final form of the effective potential for the massless scalar theory.

We have now managed to get rid of all the cut off dependece, Λ does not show in

our final expression for the effective potential, reflecting the renormalizability of

the theory.

3.1.2 Comments

In the article [28] Sindney Coleman and Eric Weinberg proposed several comments

related to this calculation.

First of all they said that their theory is renormalizable theory, as we saw in

the end of the previous section. This means that the all depence on the cutoff Λ

disappears from the final expression for the potential.

Secondly the violent infrafed singularities in the individual diagrams have be-

come singular at the origin of classical-field space. They furthermore show that

this holds for all orders in the loop expansion.

They also show that the logaritmic dependence on the coupling constant, which

is apparent in equation 3.5, disappears. They show furthermore that this holds for

higher orders. They show that the n-loop contribution to V is simply proportional

to λn+1.

They emphasize that the renormalization scale, M , is indeed an arbitrary pa-

rameter, with no effects to the physics. If different mass, M ′, is picked, then new

coupling constant would be defined as

λ′ =d4V

dφ4|M ′ = λ+

3λ2

32π2ln(M ′2

M2

).

The final form of the potential would be written as

V =λ′

4!φ4 +

λ′2φ4

256π2

(ln

φ4

M ′2 −25

6

)+O(λ3),

3.2. MASSLESS SCALAR ELECTRODYNAMICS 25

which is just reparametrization of the same function. It is just a change of defini-

tion, not a change of physics.

x

y

(a) Form of the tree level potential V0

x

y

(b) Form of the effective potential 3.11

Figure 3.3: After loop calculations potential has gained new minimums.

Lastly they note that the logarithm of a small number is negative, it appears

as though the one loop corrections have turned the minimum at the origin into a

maximum and new minimum has appeared away from origin, as shown in figure

3.3. Note that in figure 3.3, we have only plotted the form of the potential not the

actual values. That is to say that the one-loop corrections have indeed generated

spontaneous symmetry breaking. Although, in sequel they note that the apparent

new minimum occurs at value of φ determined by

λ ln〈φ〉2

M2= −32

3π2 +O(λ).

And since we expect higher orders bring higher power, the new minimum lies very

far outside the expected range of validity of the perturbative analysis.

3.2 Massless Scalar Electrodynamics

Massless scalar electrodynamics is probably the simplest example of theory that is

physically interesting and for which the effective potential has a very similar form.

Also, the final criticism of previous section is resolved: the interesting physics will

occur with full perturbative control.


3.2.1 Effective potential

In massless scalar electrodynamics our Langrangian is

L = −1

4(Fµν)

2 +1

2(∂µϕ+ eAµϕ)2 − λ

4!|ϕ|4 + counterterms,

where ϕ is complex scalar field. By substituting complex field ϕ with two real

fields ϕ = ϕ1 + iϕ2 our Lagrangian gets the form

L = −1

4(Fµν)

2 +1

2(∂µϕ1 − eAµϕ2)2 +

1

2(∂µϕ2 + eAµϕ1)2 − λ

4!(ϕ2

1 + ϕ22)2 + ct.

We now have three cases, two of them has same kind of structure as in previous

chapter 3.1 and loops are of the same form as in figure 3.2. The third comes from

the interaction within the scalar and the photons. Interaction term is form

Lint =1

2e2ϕ2

iA2µ.

And the interaction vertices are show in figure 3.4.

Figure 3.4: Interaction vertex between the scalar and the photon.

The contributing 1.loop diagrams are shown in figure 3.5, with photon running

around the polygon.

Figure 3.5: one-loop irreducible diagrams of photon [28].

We now choose to do our calculations in Landau gauge, but all physical quan-

tites are of course gauge independent. In the Landau gauge the photon propagator

is

Dµν = −ıηµν − kµkν

k2

k2 + ıε.

3.2. MASSLESS SCALAR ELECTRODYNAMICS 27

Contracting this with vertex factor we get factor in front of the loop integral. In

other words we now get three integrals of same type:

V1(α) = ı

∫d4k

(2π)4

∞∑n=1

1

2n

( αk2

)n,

where α gets different value depending from the case. For scalars α = λ2ϕ2, α = λ

6ϕ2

and for the photon α = e2ϕ2. Integral can be evaluated as we did before and we

get

V1(α) =1

24π2

[Λ2

2α +

α

4

(ln( α

Λ2

)− 1

2

)].

Loops get weights so that scalar loops are weighted by 14

and the photon loop

gets weighted by 34, because of the contraction between propagator and the vertex

factor. Thus we get effective pontetial

V = λ4!ϕ4 − B

4!ϕ4c − C

2ϕ2c

+ λΛ2

96π2ϕ2 + 3e2Λ2ϕ2

64π2 − 5λ2ϕ4

2304π2 − 3e4ϕ4

128π2

+ λ2ϕ4

256π2 ln(λϕ2

2Λ2

)+ λ2ϕ4

2304π2 ln(λϕ2

6Λ2

)+ 3e4ϕ4

64π2 ln(e2ϕ2

Λ2

).

(3.12)

Now we can use definitions 3.7 and 3.9 for solving the counterm to get the form

of the effective potential.

Solving for mass counterterm from the equation 3.7

C =Λ2

π2(λ

48+

3e2

32).

After that potential clears up little bit. The effective potential is

V =λ

4!ϕ4−B

4!ϕ4c−

5λ2ϕ4

2304π2−3e4ϕ4

128π2+λ2ϕ4

256π2ln(λϕ2

2Λ2

)+

λ2ϕ4

2304π2ln(λϕ2

6Λ2

)+

3e4ϕ4

64π2ln(e2ϕ2

Λ2

).

(3.13)

Solving for renormalized coupling constant from the equation 3.9

λ =∂4

∂ϕ4V (ϕ) = λ−B+

3λ2

32π2ln(λϕ2

8Λ2

)+

λ2

96π2ln(λϕ2

6Λ2

)+

9e4

8π2ln(e2ϕ2

Λ2

)+

55λ2

144π2+

33e4

8π2|ϕ=M ,

yields to

B =3λ2

32π2ln(λM2

2Λ2

)+

λ2

96π2ln(λM2

6Λ2

)+

9e4

8π2ln(e2M2

Λ2

)+

55λ2

144π2+

33e4

8π2.

Now we can put B back to the pontential

V = λ4!ϕ4 − λ2

256π2ϕ4 ln(λM2

2Λ2

)− λ2

2304π2ϕ4 ln(λM2

6Λ2

)− 3e4

64π2ϕ4 ln(e2M2

Λ2

)− 55λ2

3456π2ϕ4

− 11e4

64π2ϕ4 − 5λ2ϕ4

2304π2 − 3e4ϕ4

128π2 + λ2ϕ4

256π2 ln(λϕ2

2Λ2

)+ λ2ϕ4

2304π2 ln(λϕ2

6Λ2

)+ 3e4ϕ4

64π2 ln(e2ϕ2

Λ2

)= λ

4!ϕ4 + 5λ2ϕ4

1152π2 ln(ϕ2

M2

)+ 3e4ϕ4

64π2 ln(ϕ2

M2

)− 125λ2

6912π2ϕ4 − 25e4

128π2 .

(3.14)


Only task left is to simplify that expression in 3.14 and we get to our goal. This

is how we obtain the effective potential to the massless scalar electrodynamics,

V (ϕ) =λ

4!ϕ4 +

( 5λ2

1152π2+

3e4

64π2

)|ϕ|4

[ln( |ϕ|4M2

)− 25

6

]. (3.15)

As before the dependece of cut off parameter Λ has vanished.

3.2.2 Comments for Massless Scalar Electrodynamics

Let us now look little closer what we have obtained by visiting in detail on the

effective potentials.

The effective potential of the massless scalar electrodynamics 3.15 has minimum

away from the origin. For the arbitrarily small coupling constant λ, we obtain a

minimum by balancing a term of order λ against a term of order e4 ln(ϕ/M) .

Coleman an Weinberg note in their paper [28] that even though the second term

formally arises in a higher order than the first, there is no reason why λ could not

be of the same order of magnitude as e4. They say that this is actually what one

expects if we are considering the quartic scalar self-interaction as being forced on

by renormalization, to cancel the divergence in Coulomb scattering, itself of order

e4. We can now restrict ourselves in the case where λ is of the order e4 [28].

Under this assumption, the term of order λ2 in the effective potential 3.15

is so small compared to the other terms that we can neglect it completely. In

their paper Coleman and Weinberg note that to be exact we should drop it to be

consistant within ourselves [28]. Term of order λ2 would be of the same order as

the two-loop electrinagnetic corrections, which we have not computed and in this

thesis will not compute.

Because of M being completely free parameter within our theory at hand, we

can choose it to be the minimum of the potential,〈ϕ〉, keeping in mind that M has

to be within our limits of validity. Now we can rewrite the effective potential

V (ϕ) =λ

4!ϕ4 +

3e4

64π2|ϕ|4

[log( |ϕ|4〈ϕ〉2

)− 25

6

]. (3.16)

Since 〈ϕ〉 is defined to be the minimum of the potential, we can now determine

the coupling constant

0 = V ′(ϕ)|ϕ=〈ϕ〉 =(λ6− 11e4

16π2

)〈ϕ〉3

⇒ λ = 338π2 e

4.

We have now achieved redefinition of the coupling constant in terms of e4. That

means that final value of λ is independent of the initial value of λ, and therefore

independent of the initial value of M. Coleman and Weinberg note that this seems

suspicious, but in reality we have not lost any free parameters. Our theory started

3.3. COLEMAN-WEINBERG MECHANISM WITH A HIGGS PORTAL 29

with two, λ and e4, and now we have two, e4 and 〈ϕ〉. We have changed just one

dimensionless parameter to one with dimensional one [28]. They call that phe-

nomenon dimensional transmutation [28]. This dimensional transformation is an

inevitable feature of spontaneous symmetry breakdown in a massless theory. It

is stated in the paper that for fixed theory, a chage in the arbitrary renormaliza-

tion mass leads to a change in the numerical value of the dimensionless coupling

constants.

This is how Coleman and Weinberg achieved the final expression for the effec-

tive potential in the massless electrodynamics

V =3e4

64π2ϕ4(

lnϕ2

〈ϕ〉2− 1

2

), (3.17)

This funtion is parametrized only by e4 and 〈ϕ〉, all references to λ has disappeared.

If another defintion for the λ would have been adapted the final form of the effective

potential, equation 3.17, would not have been different.

3.3 Coleman-Weinberg Mechanism with a Higgs

Portal

We have now studied in lenght the mechanism presented in [28]. Now we want to

put it in use.

First we show why the scalar field used in section 3.2 cannot be the Higgs field.

After ruling out direct application of the Coleman-Weinberg mechanism to the

Standard Model, the theories that have hidden sector have gained appreciation.

The interaction term between the hidden sector and the Higgs sector is called The

Higgs portal [42,43].

We will introduce one of the models and show how it would solve the hierarchy

problem. As a guidance we use paper written by Englert, Jaeckel, Khoze and

Spannowsky [41].

3.3.1 Why not the Higgs?

The Higgs potential is a part of the Standard Model Lagrangian

V (H) = µ2SMH

†H +λH2

(H†H)2, (3.18)

where µ2SM < 0 has been taken into account. Higgs potential is invariant under

U(1) transformations .

H → eıαH

Now we can choose the H to be H = 1√2eıαv and find the minimum of potential

with respect to v. The potential is

V (v) =1

2µ2SMv

2 +λH8v4.


The derivative of the potential

∂

∂vV (v) = µ2

SMv +λH2v3 = 0.

From this we can find the minimum v2 = −2µ2MS

λH, mass of the Higgs boson, m2

h =∂2

∂H2V (H)|H=0 = λHv2. From here we can solve for the Higgs mass parameter

⇒ µ2SM = −1

2λHv

2 = −1

2m2h. (3.19)

This value for the Higg mass parameter fits to the experimental data that an

expectation value for the Higgs field v ' 246 GeV and the Higgs mass mh ∼ 125

GeV [41].

We start from where we ended up in section 3.2.2. Effective potential in the

vacuum was

V =3e4

64π2ϕ4(

lnϕ2

〈ϕ〉2− 1

2

). (3.20)

After the shifting of the field the mass of the scalar can be reformed [28] by

m2S = V ′′(〈φ〉) =

3e4

8π2〈φ〉2. (3.21)

The would-be Goldstone boson combines with the photon to make a massive vactor

meson. Its mass is given by

m2V = e2〈φ〉2. (3.22)

This means that the Higgs boson mass has an upper limit in the theory

m2S =

3e2

8π2m2V < m2

V .

In the Standard Model Higgs boson mass is experimentally found to be around

125 GeV [7, 8]. Corresponding vector mesons on the other hand are lighter, W-

bosons only 80 GeV and Z-boson 91 GeV. If this theory would describe our Uni-

verse the Higgs mass would be too light. This now means that the scalar field

that we have calculated the effective potential for, cannot be the Higgs field. In

the next section we consider more complex case in which the Higgs and the vector

bosons are free to take their observed values.


3.3.2 Higgs Portal

To resolve this problem we follow the development presented in ref. [41]. In that

paper authors introduce simple solution, where regular Standard Model is extended

by scalar field that is in the hidden sector, that means that scalar could be a dark

matter candidate. That scalar field interacts with the Higgs field through the

Higgs portal [42,43]. That way the Higgs can take its observed mass.

The classical potential is of the form,

Vcl(φ,H) =λH2

(H†H)2 − λP (H†H)|φ|2 +λφ4!|φ|4, (3.23)

where λP interaction coupling constant, when λP → 0 particles decouple. First

term here is the Higgs field self-interaction and the last is the scalar field self-

interaction, λH and λφ are respectively their coupling constants. The second term

is the Higgs portal, that couples the hidden sector to the Standard Model.

Potential is stable if V (H,φ) > 0. To check this we complete the square by

adding ±(λPλH

)2

|φ|4 to the potential,

⇒ Vcl(φ,H) =λH2

(H†H − λP

λH|φ|2)2

+1

24λH(λφλH − 12λ2

P ). (3.24)

The potential is stable as long as

λφλH > 12λ2P .

By comparing potential 3.23 to Higgs potential 3.18, we can see that for non-

vanishing λP Higgs mass parameter can be generated as µ2SM = −λP 〈|φ|〉2.

In the paper [41] authors note that in the potential 3.23 there is no mass terms,

so we have completely scale-free potential. We can now employ the Coleman-

Weinberg megnaism in the Higgs portal theory. The Coleman- Weinberg mecha-

nism acts on the hidden sector, so it only affects the complex scalar field φ. Now

we can substitude the self-interaction of the scalar field in 3.23 by effective poten-

tial, V (φ), from the equation 3.16. So the effective potential for the Higgs portal

model would read

V (φ,H) =λφ4!|φ|4 +

3e4φ64π2 |φ|4

[ln(|φ|2〈|φ|〉2

)− 25

6

]−λP (H†H)|φ|2 + λH

2(H†H)2.

(3.25)

The goal here is to show that the this potential now really breaks the ele-

crtoweak symmetry and generates mass scale in which we can now achieve the

observed mass of the Higgs boson.

The easiest way to visualise this is to conside a near decoupling limit [41].

If λP 1 we can see the process of symmetry breaking independently in the


two different sectors and electroweak symmetry breaking effectively as a two step

process [41]. In the first step the Coleman-Weinberg mechanism creates vacuum

expectational value for the scalar field through dimensional transmutation. In the

second step the vacuum expectational value is transmitted to the Standard Model

through Higgs portal creating an effective mass parameter for the Higgs

µ2SM = −λP 〈|φ|〉2.

The equation 3.19 dictates that µ2SM fixes the electroweak scale, specially

−µ2SM =

1

2m2h =

1

2(125 GeV)2 and − µ2

SM =1

2λHv

2 ≡ λH〈|H|〉2.

We can now put these together and find, how the vacuum expectational values

depend on each other

〈|φ|〉2 =1

λP

1

2(125 GeV)2 =

λHλP〈|H|〉2. (3.26)

We must now convince ourselves in that the dimensional transmutation hap-

pens in the same way as it happens in the section 3.2. The dimensional trans-

mutation is inevitable to get rid of the λφ- term of the effective potential, and to

verify we can use the potential as it is written in the equation 3.25.

We can now calculate the minimum of the field

∂

∂φV (φ,H) =

1

6

(λφ−

33

8π2

)〈φ〉3−2λP 〈|H|〉2〈φ〉 =

1

6

(λφ−

33

8π2e4φ−12

λ2P

λH

)〈|φ|〉3 = 0.

From here we can solve for λφ,

λφ =33

8π2e4φ + 12

λ2P

λH' 33

8π2e4φ. (3.27)

At the limit of weak coupling λP this is just a little deviation from the massless

scalar electrodynamics as shown in section 3.2.2. So now we have shown that the

dimensional transmutation happens, in more complex theory as it happens in the

massless scalar electrodynamics, thus the effective potential can be used as it is in

the 3.25. And the hidden scalar particle gains a mass m2s =

3e4φ16π2λp

(125GeV)2 and

the Higgs boson is free to take mass of 125 GeV.

Difference to previous section 3.3.1 is that the symmetry break through Coleman-

Weinberg mechanism happens in the hidden sector, that means Standard Model

particles do not participate to the effective potential 3.25. That means that while

in the hidden sector the scalar can be lighter than the corresponding hidden sector

vector bosons. What is crucial here that massive vector bosons of the Standard

Model do not have role in Coleman-Weinberg mechanism as they did in the pre-

vious section 3.3.1. After the symmetry break they gain their weights via Higgs

mechanism as they do in Standard Model.


3.3.3 Phenomenology of the Higgs Portal Model

In the hidden sector we now have the field, φ, and the gauge field, Xµ. After

φ acquires a non-vanishing vacuum expectational value, the gauge field acquires

mass

mX = eφ〈φ〉. (3.28)

And the scalar bosons mass can be expressed as

mϕ =3e4

φ

8π2〈φ〉2 =

3e2φ

8π2m2X . (3.29)

The dominant interaction between the hidden sector and the Standard Model

is via the Higgs portal coupling λP . The lowest order effect arises from the mixing

of the Standard Model Higgs, HT (x) = 1√2(0, v + h(x)) and the hidden Higgs,

φ = 〈φ〉 + ϕ. This provides additional decay channel for the Higgs boson, this

could be detected as missing energy in the LHC experiments. The two scalars, h

and φ, mix via mass matrix,

m2 =

(m2h + ∆m2

h,SM −κm2h

−κm2h m2

ϕ + κ2m2h

), (3.30)

where κ is the mixing parameter,

κ =

√2λPλH

, (3.31)

and the Higgs mass

m2h = λHv

2. (3.32)

These masses for the hidden scalar 3.29 and the Higgs 3.32 are as they were on

the decoupled case, where λP = 0.

∆m2h,SM is the one loop correction the Standard Model Higgs mass,

∆m2h,SM =

1

16π2

1

v2(6m4

W + 3m4Z +m2

h − 24m4t ) ' −2200GeV2. (3.33)

We have introduced the one loop corrections for the Standard Model side of sector

to be consistent as we have introduced one loop corrections in the hidden sector

as well. Numerically, these corrections are dominated by the top-quark loop and

therefore negatiove. While these corrections lead to small contribution in the limit

of small λP and at large m2ϕ, they lead to interesting effects for the case of small

m2ϕ and moderate Higgs portal coupling.

This matrix can be easily digonalized with rotation,(h1

h2

)=

(cos θ sin θ

− sin θ cos θ

)(h

ϕ

), (3.34)


where the mixing angle is given in the limit of small mixing as

θ ' κm2h

m2ϕ −m2

h −∆m2h,SM

1. (3.35)

Up to order θ2 and to the leading order in λP the masses of the two eigenstates

are

m2h1

= (m2h + ∆m2

h,SM)(1 +O(θ2)), m2h2

= m2ϕ(1 +O(θ2)). (3.36)

Fixing the dominantly Standard Model like state h1 to have a mass of 125 GeV

we are left with two remaining parameters: the mixing angle θ and the mass of the

second eigenstate. Now we can look at possible constraints to the parameters that

follow from transmitting the electroweak symmetry break to the visible sector.

The Standard Model Higgs self-coupling is fixed by the ratio of known elec-

troweak scales, while the hidden scalar self-coupling is determined from the Coleman-

Weinberg dimensional transmutation condition,

λH =(mh

v

)2

' 1

22=

1

4, λφ =

33

8π2e4φ. (3.37)

In the hidden sector there are two undetermined parameters: the hidden sector

gauge coupling e2φ, and the small portal coupling λP . In this casem the two mass

scales associated with the hidden scalar are fixed,

〈|φ|2〉 =1

2λPm2h, m2

ϕ =3e4

φ

8π2〈|φ|2〉 =

3e4φ

16π2

1

λPm2h, (3.38)

and the hidden sector vector mass is given by

m2X =

8π2

3e2φ

m2ϕ =

e2φ

2λPm2h. (3.39)

Alternatively, the two free parameters can be chosen to be the mass of the

hidden Higgs, m2ϕ, and the Higgs portal coupling, λP . In this case, gauge coupling,

e2φ, and the mixing parameter are determined via

e2φ = 4π

√λP3

mϕ

mh

, κ =

√2λPλH

. (3.40)

In analogy to the Standard Model case gauge coupling can be expected to be

of order 0.1 - 1 but it could be much smaller. In the case of small gauge coupling

one should explain incredibly small value of λφ.More importantly, for small m2ϕ it

is crucial to take into account higher order corrections to 3.36. So that

m2h2

= m2ϕ(1 +O(θ2)) + θ2

∆m2h,SM

m2h

(m2h + ∆m2

h,SM). (3.41)


As the Standard Model corrections to the Higgs mass is negative we must require

that physical mass in equation 3.41 stays positive. From that requirement we

obtain minimal value for m2ϕ

m2ϕ ≥ m2

ϕ,min =2λPλH

( |∆m2h,SM |

m2h + ∆m2

h,SM

)m2h. (3.42)

The minimal value of m2ϕ translates into minimal value for

e2φ ≥

2λPλh

√λH

8π2

3

|∆m2h,SM |

m2h + ∆m2

h,SM

, (3.43)

which in turn translates into the lower bound for the hidden U(1) gauge boson,

m2X ≥ m2

h

√1

λH

8π2

3

|∆m2h,SM |

m2h + ∆m2

h,SM

. (3.44)

For the physical massm2h2

equations 3.41 and 3.42 entail that values much below

m2ϕ,min require some amount of fine-tuning as it involves a cancellation between m2

ϕ

and the Standard Model correction. From now on we can therefore consentrate

on the case of moderate gauge coupling and hidden higgses with masses mϕ ≥MeV [41].

If the mass mh1 > 2mh2 the Standard Model Higgs can decay into two hidden

Higgses. In leading order in the mixing angle this decay occurs via the term

Ld ∼ −λPvhϕ2,

and the Standard Model -like Higgs trilinear interaction. The rotation to the phys-

ical mass eigenstates h1 and h2 forces the trilinear couplings to be more involved.

For now we settle for the case of small mixing h2 ∼ ϕ. Hence the dominant part

of the corresponding partial decay width is

Γh1→h2h2 =4λ2

Pv2

16π

√m2h1− 4m2

h2

m2h1

, (3.45)

and needs to be taken into account for the Higgs modified branching rations.

Similar equation holds formh2 > 2mh1 , with v changed to√

2〈|φ|〉 andmh1 changed

with mh2 . In this setup there are no light hidden sector particles into which

the hidden Higgs can decay. The h2 therefore decays back into Standard Model

particles via the mixing with the Higgs and its couplings to light particles [41].

The branching ratios are the same as for the Standard Model Higgs with mass

m2h2

, but the width, as well as the production cross sections from visible matter,

are reduced by a factor sin2 θ,

Γh2→XXc = sin2 θ ΓSMh→XXc(mh = mh2),

σ(XY → h2) = sin2 θ σSMXY→h(mh = mh2).


Note, that already the SM Higgs decay width is quite small, ΓSM(mh ' 125GeV) '4 MeV and decreases more or less linearly with the mass [41]. Combining this with

a small mixing angle, h2 becomes an extremely narrow resonance.

Figure 3.6: Scatter plot of model at hand for 105 randomly generated parameter

choices in the (λP ,mh2) plane. Points below the black dash-dotted line require

some fine-tuning. The region excluded by current LHC measurements is shown

in red. The cyan region can be probed by LHC with high luminosity and the

orange region shows a projection for a combination of a high luminosity LHC

with a linear collider. Light blue indicates constraints from stellar evolution. The

constraints on the parameter space for a Landau pole separation of 4, and 16 orders

of magnitude are included in yellow and light green, respectively. The remaining

allowed parameter points are depicted in green [41].

In figure 3.6 is shown the result of a parameter scan projected on the (λP ,mh2)

plane, done in ref. [41], mh1 has been identified to 125 GeV. A constraint can

be imposed from theoretical reasoning. From equation 3.27 we can see that λφgrows as eφ and/or λP are increased raising the possibility of a nearby Landau

Pole. Requiring that there is no Landau pole in λφ for at least a few orders of

magnitude puts already fairly strict limits on both λφ and λP . Neglecting the

λ2P contribution to the running of λP the solutions to the renormalization group

equations are given in ref. [28]. In figure 3.6 is show the constraints arising from

a hierarchy of 4 and 16 orders of magnitude between 〈φ〉 and the Landau pole

yellow and light green, respectively. We can see that this automatically restricts

us to fairly small λP . The used approximation is conservative in the sense that the

λ2P contribution to the running of λφ is positive speeding up the approach to the


Landau pole. On the other hand for small λP the neglected term quickly becomes

very small [41].

A significant decay of the Higgs candidate into fermions is yet to be measured.

Current constraints on h→ b~b follow from biasing the coupling fit with the Stan-

dard Model assumption of a total Standard Model -like Higgs decay width. The

observed rates, at the current precision can be understood as a limit on the total

Higgs width itself. Given that we have a potentially large coupling to a new decay

channel at a large available phase space equation 3.45 an upper limit on the total

Higgs width constrains the model. Recent analyses suggest Γh/4 MeV ≤ 1.3 and

that bound has been included in the scan [41]. The scan done in ref. [41] authors

also display the improvement of the ruled-out region due to the combination of a

high-luminosity LHC run in combination with a linear collider on the basis of the

most recent coupling fits of ref. [44]. Note that, other than at a hadron collider,

the total Higgs width can be measured by correlating Higgs production in weak

boson fusion e+e− → ν~νh and the decay h→ WW at the Γh/4 MeV ≤ 10% level.

From figure 3.6 it can be seen that there is a large parameter region of the

model allowed by current measurements. The allowed region, of course further

extends to smaller λP and also to larger masses. The model can be efficiently

constrained by measuring the Higgs candidates cross section and decay width as

precisely as possible, which can be done extraordinarily well at a precision collider

instrument such as a future linear collider [41]. The small funnel region at around

mh2 ' mh = 125 GeV follows from relaxed bounds and kinematic suppersion in

the vicinity of the Higgs candiadate. mh2 within this range is then unconstrained

more or less irrespective of the precise value of λP .

Since, λP is small by consistency and renormalization group arguments, we

face small mixing with the hidden sector which effectively yields a phenomeno-

logically decoupled Higgs partner in the single Higgs channels when background

uncertainties are taken into account. When the mixing is rather larger sensitivity

in Standard Model-like Higgs searches can provide powerful means to constrain the

model for heavy mh2 . Given the small width, standard analyses can be straight-

forwardly extended beyond the current upper limit of mh1 ≤ 1 TeV [41].

The suppression of single-h2 phenomenology can in principle be counteracted

in the di-Higgs channels pp → h2 → h1h1 → Standard Model. The resonance is

extremely narrow, and for the parameter space mh2 > 2mh1 it naturally appears

in the TeV regime. While the small mixing angle naively means a suppressed

s-channel contribution of h2 to the di-Higgs phenomenology [41], it exclusively

decays to a Standard Model -like di-Higgs system in this setting with a potentially

large coupling ∼ λPv ∼ 1 GeV. The small coupling of h2 to the top quarks running

in the gluon fusion loops however can typically not be beaten by the h2h1h1 vertex

[41]. This contribution has to be put in contrast to the off-shell h1h1h1 vertex

∼ v λP 〈|φ|〉 which is Standard Model -like and, more importantly for high

energetic Higgses, to the box-induced continuum gg → h1h1 production. In their


paper authors of ref. [41] have performed a full one-loop computation of pp →h1h1 → visible via gluon fusion in the this model and have scanned the cross

section for a couple of parameter points and always find a di-Higgs cross section

of O(16) fb.

In total, precision analyses of the Higgs-like candidate at 125 GeV and extend-

ing Higgs boson-like searches beyond 1 TeV therefore provide the best handles to

constrain this model in its simplest implementation. The portal parameter, which

is required to be small in the limit of light h2 can be efficiently constrained by

measuring the h1 couplings at a future linear collider. Excluding heavy h2 fields

in high luminosity LHC searches limit the parameter space for λP ≤ 0.001 [41].

Low energy measurements on the other hand are highly sensitive to very light

masses, for example fifth force measurements can probe mixing angles sin θ < 10−10

for mh2 ≤ 10−2, which limits the model for such very small (λP ,mh2) combinations.

For moderate masses mh2 ≤ 100 keV stellar evolution sets strong constraints on

scalar couplings to two photons. The coupling of h2 to two photons is given by,

gh2γγ = sin θgSMhγγ ,

these bounds can be translated into limit on sin θ ≤ 10−3.86 for masses mh2 ≤ 100

keV [41].

The model’s phenomenology is that of a Higgs portal model, however with

constraints imposed that arise from generating the electroweak scale via a small

visible-hidden sector coupling [41]. The modifications compared to the Standard

Model are generically small, and exclusion bounds are driven by precision investi-

gations of the Higgs boson candidate. In essence, electroweak symmetry breaking

proceeds along the lines of the Standard Model, with modifications only due to

small mixing effects and total Higgs width modifications. All these quantities can

be determined most precisely at a future linear collider [41].

Chapter 4

Thermal Quantum Field Theory

In this chapter we briefly discuss the formalism of the thermal quantum field

theory.

4.1 Reminder of Statistical Physics

In statistical physics the basic quantity to compute is the partition funtion, Z.

We can take our quantum mechanical system to a finite temperature T and com-

pute the partition function, from which we can calculate all physical quantities in

equlibrium. We can employ the canonical ensemble [45]. The partition function is

defiend

Z ≡ Tr(e−βH), where β =1

T.

From the partition function we can now define, for example, free energy F, the

entropy S and the avarage energy E:

F = −T lnZ,

S = ∂∂TF = lnZ + 1

TZTr(He−βH) = −FT

+ ET,

E = 1ZTr(He−βH),

where H is the Hamiltonian of the system.

4.2 Path Integral for the Partition Function

We want to calculate the partition function and in a thermal quantum field theory

it is done by considering path integrals. We go ahead and introduce this procedure

for free scalar fields. It is just an introduction and reader who wants to learn more

should consult books of thermal field theory, such as ref [46], ref [47] and ref [48].

39

40 CHAPTER 4. THERMAL QUANTUM FIELD THEORY

4.2.1 Free scalar fields

We start by considering the Lagrangian of the scalar field in d-dimensions

L =1

2(∂tφ)2 − 1

2(∂iφ)(∂iφ)− V (φ).

formally the scalar field theory is nothing but collection of almost independent

harmonic oscillators, one at every point in space [45]. These oscillators interact

through a derivate term 12(∂iφ)(∂iφ) wich in the statistical physics means that the

oscillator is coupled to its nearest neighbours,

∂i 'φ(t, ~x+ εei)− φ(t, ~x)

ε,

where ei is unit vector in the direction i. The path integral form of the partition

function then is

Z = C

∫φ(β,~x)=φ(0,~x)

Dφ(τ, ~x)e−∫ β0 dτ

∫dd~xLE ,

where

LE = −L(t→ −ıτ) =1

2∂µφ∂µφ+ V (φ)

is the euclidian Lagrangian. We can now rewrite the path integral in Fourier

representation. Then the dependence on τ can be expanded as

φ(τ, ~x) = T

∞∑n=−∞

φ(ωn, ~x)eıωnτ , ωn = 2πTn.

For the space coordinates, it is useful to make each direction finite for the

moment, of extent Li, an impose periodic boundary conditions. After that we

take the limit where Li →∞, thus we get that Fourier expansion of φ is

φ(τ, ~x) = T∑ωn

1

V

∑~k

φ(ωn, ~k)eıωnτ−ı~k·~x.

at free case, where V (φ) = 12m2φ2,

−∫ β

0

dτ

∫dd~xLE = −1

2T∑ωn

1

V

∑~k

(ω2n + ~k2 +m2)|φ(ωn, ~k)|2,

and denoting Ek =√~k2 +m2, we thus get that

Z =∏~k

[T∏n

(ω2n + E2

k)−1/2

∏n′

(ω2n)−1/2] =

∏~k

e− 1T

(Ek2

+ln

(1−e−βEk

))

Now that we have introduced the path integral formalism, we turn to the calcula-

tion of the polarization tensor at finite temperature.

4.3. CALCULATION OF POLARIZATION TENSOR 41

4.3 Calculation of Polarization Tensor

We now introduce the calculation of polarization tensor π(ωn, ~p) in massless case.

Later in our discussion we calculate the polarization tensor in massive case and

we closely follow these calculations.

For a massless theory the dominant contribution to the polarizaton tensor at

order λN gomes from the infrared-divergent diagram, graphically, this corresponds

to a loop with one zero mode line, dressed with N non-zero mode bubbles shown

in figure 4.1. The small bubbles are one-loop polarization ternsors in the infraded

limit:

π(1)(ωn, ~k) = π(1)(0) = 3λT∑n

∫d3~q

(2π)3

1

ω2n + ~q2

= λT 2

4

Figure 4.1: infrared-divergent diagram

In this massless limit we obtain the infrafed limit by setting the zeroth com-

ponent of the external momentum to zero and taking the limit that the spatial

components approach zero

π(1)(p0 = 0, ~p→ 0) ≡ π(1)(0).

The infrared limit of the one-loop polarization tensor is not dependent of the

momentum and the sum over N can be done explicitly [49]. The result here is an

expression for the polarization tensor π(ωn, ~p) as a fuction of the propagator which

now has the effective mass given by the square root of π(1)(0).

π(ωn, ~k) = π(0) = 3λT∑n

∫d3~p

(2π)3

1

ω2n + ~p2 + π(1)(0)

(4.1)

Finally the donminant contribution comes from the infrared limit of this result

4.1.

Chapter 5

Effective Potential at Finite

Temperature

In their article Macroscopic consequences of the Weinberg model Kirzhnits and

Linde observe that spontetneously broken symmetries are usually restored at high

temperature [50]. The nature of the symmetry-restoring phase transition is deter-

mined by the behavior of the effective potential. The thermal fluctuations of the

field to the leading order will eventually dominate, when temperature is increased

and symmetry will be restored.

In particular intrest is in the electroweak phase transition, because this infor-

mation is capable of explaining the observed net baryon number in the Universe

in the terms of the baryon-number violation in the electroweak theory and cosmo-

logical models of baryogenesis [49].

In this chapter we are going through calculation of the effective potential in

the finite temperature field theory. We are using as a guild the article Effective

potential at finite temperature in the standard model written by M. E. Carrington.

In her paper [49] Carrington calculates the effective potential in finite temperature

effective potential for the scalar field theory and for the whole Standard Model. We

consider first the case of scalar fields, then the Standard Model without fermions

and lastly scalar field with U(1) gauge field.

5.1 The Scalar Theory

Let us now go back to the simplest of our cases. This time we are considering

Lagrangian that is not scale invariant meaning that the mass term is present in

the Lagrangian. The scalar case acts as a guideline for the discussions in the next

sections.

42

5.1. THE SCALAR THEORY 43

5.1.1 Calculation of the effective potential

In the scalar theory the Lagrangian density is

L =1

2(∂µφ)2 +

1

2c2φ− 1

4λφ4 + counterterms. (5.1)

The Lagrangian is symmetric under the trasformation φ → −φ. However the

vacuum state is not. The vacuum has a degeneracy that leads to spontatneous

symmetry breaking. We can now express the field φ as a sum of the constant v

and a field χ,

φ = v + χ.

After this shift the mass of the field χ is m2(v) = 3λv2 − c2, and the Lagrangian

does not have the original symmetry [49].

As stated above we will calculate the effective potetial following paper [49] and

extremize it respect to v to find the equilibrium value 〈v〉. If this equilibrium goes

to zero at some finite themperature, then the symmetry has a phase transition

between broken- and full-symmetry phases.

To the lowest order we have the potential

V0(v) = −1

2c2v2 +

1

4λv4. (5.2)

At tree level the potential gives classical minimum 〈v〉0 = cλ1/2

which yields to the

classical mass

m2(〈v〉0) = V ′′(v)|v=〈v〉0 = 3λ〈v〉20 − c2 = 2λ〈v〉20 = 2c2.

To the next order in the loop expansions there is only one diagram, shown in fig

Figure 5.1: Only loop that contirbutes at this level of expansion.

5.1, that contributes to the potential. The potential gets the form

V1(v) =1

2(2π)4

∫d4k ln[k2

E +m2(v)]. (5.3)

At finite temperature, we rewrite the integral with respect to time as a sum over

frequencies as we did when deriving the propagator in the section 4:∫dτ

2π→ T

∑n

f(τ = ıωn), ωn = 2πnT,

44 CHAPTER 5. EFFECTIVE POTENTIAL AT FINITE TEMPERATURE

As a result the integral 5.3 splits to zero-temperature loop correction term, V(0)

1 (v)

and the finite-temperature loop correction term, V(T )

1 (v). Integral now has the

form

V1(v) =T

2

∑n

∫d3~k ln(ωn + (~k2 +m2(v))).

Let us now forget the 3-dimensional integral and take a look at the function

F (Ω) =T

2

∑n

ln(ω2n + Ω2),

where Ω =

√~k +m2(v). Taking first derivate of the F (Ω) we find

F ′(Ω) = T∑n

Ω

Ω2 + ω2n

=∑n

Ω/T

(Ω/T )2 + (2πn)2=

1

2coth

( Ω

2T

).

Now we can integrate it back to F (Ω) and drop the integration constant [51]. Thus

we obtain

F (Ω) = T ln[2 sinh

( Ω

2T

)]=

Ω

2+ T ln(1− e−Ω/T ).

From this form we can now pick the first term to be the zero-temperature contri-

bution and second term to be the finite-temperature contribution [51].

The zero-temperature part is now

V(0)

1 (v) =1

2(2π)3

∫d3~k[~k2 +m2(v)]1/2, (5.4)

it represents the shift in the vacuum energy from zeropoint oscillations of the field.

The integral is divergent and it is cutoff at Λ.

As previous cases the counterterms contribute at one-loop level. This the-

ory can be renormalized at zero-temperature by introducing mass counterterm,

coupling constant counterterm and a costant that can be used to cancel the v-

independent part of the vacuum energy. The counterterms are of the form

Lcounterterms =A

2v2 +

B

4v4 + C.

The counterterms A and B can be solved by requiring that the position of the

minimum and mass remain at their classical values[dV (0)(v)

dv

]v=〈v〉0

= 0,[d2V (0)(v)

dv2

]v=〈v〉0

= 2c2,

where V (0)(v) = V0(v) + V(0)

1 (v). For now we leave C untouched, in the end it

is used to counter terms that do not depend on v. Thus we obtain part of the

effective potential

V (0)(v) = −c2

2v2 +

λ

4v4 +

1

64π2m4(v) ln(

m2(v)

2c2) +

21λc2

64π2v2 − 27λ2

128π2v4. (5.5)


Finite teperature part of the loop corrections

V(T )

1 (v) =T

2π2

∫dk k2 ln(1− e−

√k2+m2(v)/T ), (5.6)

For limit m(v)/T 1, this integral can be expanded [49] as of powers of m(v)/T

V(T )

1 (v) = T 4[− π

2

90+m2(v)

24T 2−m

3(v)

12πT 3− m4(v)

32π2T 4ln(m(v)

4πT

)+O(m4(v)/T 4)

]. (5.7)

From these potentials 5.5 and 5.7 we can see that terms that are proportional

m4(v) ln(m(v)) cancel each other.

If we stop here and extremize the effective potetial with respect of v we obtain

the mean-field value 〈v〉.

5.1.2 The Ring Diagram Contribution

Next order correction does not come from the two-loop term but from the ring

diagrams in figure 5.2, which are of the order λ32 . It is known that in the massless

Figure 5.2: The ring diagrams [49]

limit of the scalar field theory at finite temperature ring diagrams give contribu-

tions [45], therefore we must consider that in this case the order λ32 contributions

are meaningful at the limit of small m(v)/T [49].

Physically ring diagrams give contributions from long distance effects. These

effects can be obtained by going beyond the mean-field approximation at one-loop

level.

Previously we have shifted the one-point function φ by c-number v. To go

beyond mean-field level, we need to consider effective potential as a function of

both v and the two-point function π(ωn, ~p). Afterwards we can extremize the

potential with respect to v and π(ωn, ~p).

At the one loop level now we get

V (v) = V0(v) +T

2

∑n

∫d3~p

(2π)3ln[ω2

n + ~p2 +m2(v) + 〈π(ωn, ~p)〉T 2

]. (5.8)

〈π(ωn, ~p)〉 is to be obtained self-consistently as the choise of π(ωn, ~p) extremizes

the effective potential. This can be obtained within perturbation theory.

In this case the field is not massless, and we can expect to get contribution

from the ring diagrams. Proceeding from here analogously to the massless case,


the only difference is that we have added the mass. The one-loop polarization

tensor is given by

π(1)(ωn, ~k) = π(1)(0) = 3λT∑n

∫d3~q

(2π)3

1

ω2n + ~q2 +m2(v)

= λT 2

4

[1 +O

(m(v)

T

)]And the sum over N now gives

π(ωn, ~k) = π(0) = 3λT∑n

∫d3~p

(2π)3

1

ω2n + ~p2 +m2(v) + π(1)(0)

. (5.9)

Now we have obtained the polarization tensor in terms of the propagator, with the

effective mass√m2(v) + π(1)(0). Expanding this now in terms of m(v)

Twe obtain

π(0) = λT 2

4

[1 +O

(m(v)

T

)]' λ

T 2

4. (5.10)

Terms of the order O(m(v)/T ) can now be dropped, because we are at the limit

where m(v)/T 1. Only limit where these contributions would have effect is

m(v)/T ∼ 1, but in that limit π(1)(0) ∼ λm2(v), which is small compared to

m2(v). The zeroth term of polarization tensor can be used because it is reasonable

good approximation at the regime in which, the ring diagrams give contribution.

The whole effective potential is now

V (v) = V0(v) +T

2

∑n

∫d3~p

(2π)3ln[ω2

n + ~p2 +m2(v) + λT2

4

T 2

]. (5.11)

This calculation has now produced for us potential which is sum of three parts.

First is the tree level which is marked explicitly, the second is the loop calculation

which we did in two parts zero-temperture part and the finite-temperature part, the

third is the ring diagram contribution. Let us denote ω2n + ~p2 +m2(v) ≡ f(n, p, v),

then

T2

∑n

∫d3~p

(2π)3ln[f(n,p,v)+λT

2

4

T 2

]= T

2

∑n

∫d3~p

(2π)3ln[f(n,p,v)T 2 (1 + λT

2

4f−1(n, p, v))

]= T

2

∑n

∫d3~p

(2π)3ln[f(n,p,v)T 2

]+ T

2

∑n

∫d3~p

(2π)3ln[(1− (−1)λT

2

4f−1(n, p, v))

]= V1(v) + T

2

∑n

∫d3~p

(2π)3

∑N

1N

[− λT 2

4f−1(n, p, v)

]N= V1(v) + Vring(v).

After doing the integration on the ring contibution we get [49]

Vring(v) = − T

12π

[(m2(v) + π(0))3/2 −m3(v)

].


Now we have all the pieces we need for our effective potenial:

Veff = V (0)(v) + V(T )

1 (v) + Vring(v)

= −12c2v2 + 1

4λv4 + 21λc2

64π2 v2 − 27λ2

128π2v4

+m2(v)24

T 2 − T12π

(m2(v) + λT2

4)3/2 +O(m4(v))

(5.12)

From the final expression of the effective potential terms that are independent of

v have been canceled by counterterm C and terms proportional to m4(v) ln(m(v))

have canceled each other as well as the terms proportional to m3(v).

5.1.3 Conclusions

The cancellation of the terms proportional to m3(v) is especially important since

for the m3(v) = 3λv2 − c2 < 0, those terms would be imaginary in the finite-

temperature part of the one loop contribution.

Final effective potential is real if temperature, T , is large enough. Specifically

m2(v) + λT 2

4= 3λv2 − c2 + λ

T 2

4≥ 0,

this condition is satisfied for all v if

T ≥ T1 =2c

λ1/2= 2〈v〉0.

For large enough temeperature, T > T1 effective mass is real, to the lowest order

m2eff =

∂2

∂v2Veff |v=0 = −c2 +

λ

4T 2.

At the limit where effective mass goes to zero at the lowest order, T = T1, and

small v the the term −T1(3λ)3/2

12πv3 dominates. At the large v the quartic term in

the tree potential dominates.

v

V (v)

Figure 5.3: Form of the potential at T = T1.


Thus the effective potential gets the form shown in the figure 5.3. When the

temperature is increased, the coefficent before the quadratic term flips sign and

the curvature at the origin becomes positive. When that happens the effective

potential gains a local minimum at the origin. That indicates that the system has

a first order phase transition at some temperature [49] and symmetry is therefore

regained. That is indeed the case and it has been shown numerically that first

order phase transition occurs [52] by K. Takahashi.

5.2 The Standard Model Without Fermions

Throughout this thesis we have made conscious choice not to introduce fermions,

we will keep this in mind for this next chapter. In this chapeter we will calculate

the effective potential for Standard Model rejecting any fermions.

5.2.1 The Lagrangian Density

The Lagrangian density for this case

L = LGauge + LHiggs + Lfixing,

contains several different parts. These parts are all invariant under SU(2)×U(1)Ytransformations. Lets now take a closer look at each of these parts.

The gauge fied part is

LGauge = −1

4F aµνF

µνa −

1

4FµνF

µν .

Where F µν is field strength tensor. First term describes SU(2)-gauge fields and a

runs from 1 to 3.

The Higgs part is

LHiggs = (DµH)†(DµH) + c2(H†H)− λ(H†H)2,

describes complex Higgs doublet. The covariant derivative is

Dµ = ∂µ +ı

2g′Bµ +

ıσa

2gAaµ,

where σa are Pauli matrices. We define AµA = (Aµa , Bµ). The field descibe above

are eigenstates of isospin and hypercharge and thus Lagrangian is symmetric under

SU(2)×U(1)Y transformation.

The gauge fixing part of the lagrangian is

Lfixing = − 1

2ξ(∂µAaµ −

1

2ξgvχa)2 − 1

2ξ(∂µBµ −

1

2ξg′vχa)2,

5.2. THE STANDARD MODEL WITHOUT FERMIONS 49

where χa = φ2, φ3, φ4. Because we choose to work in the Landau gauge the gauge-

fixing parameter ξ → 0. The crossterms in the gauge fixing part combined with

the cross terms from the term (DµH)†(DµH) produces the total diverfences, which

integrate to zero. In the Landau gauge, the ghost field are massless and do not

contribute to the v-dependent form of the one-loop effective potential, so we do

not need to worry about those. The effective potential itself is dependent on the

choice of the gauge, however all physical quantities calculated from the effective

potential, such as the critical temperature, are gauge independent [53].

At the original basis where the relevant fields were AµA, the vector propagator

is

ıDABµν (q) = −ı

[ ηµν − qµqν/q2

q2 −M2(v) + ıε

]AB,

where M2(v) is the mass matrix defined at equation 1.2,

M2(v) =

g2

4v2 0 0 0

0 g2

4v2 0 0

0 0 g2

4v2 −gg′

4v2

0 0 −gg′

4v2 g′2

4v2

.

After that matrix is diagonalized and we have gained the fields W±µ , Zµ and Aµ

we obtain the propagators.

ıDµν(q) = −ı[ ηµν − qµqν/q2

q2 −m2(v) + ıε

], (5.13)

for s W±µ and Zµ, where m2(v) = m2

W (v) or m2Z(v). And for the field Aµ the

propagator

ıDµν(q) = −ı[ηµν − qµqν/q2

q2 + ıε

]. (5.14)

From here we move on to calculate the effective potential.

5.2.2 Zero-temperature Part of the Effective Potential

We can calculate the effective potential in the same manner that we did in in

previous section. At the tree level we have

V0(v) = − c2v2 +

λ

4v4. (5.15)

To calculate next order in the loop expansion we need to consider the one-loop

diagrams and the ring diagrams.

The one loop contributions contain the zero-temperature and the finite-teperature

parts. We are now considering one loop contributions from the Higgs boson and


gauge boson loops. So that

V1(v) = V(0)

1 (v) + V(T )

1 (v),

V(0)

1 (v) = V(0)

1,H(v) + V(0)

1,gb(v),

V(T )

1 (v) = V(T )

1,H (v) + V(0)

1,gb(v).

We can write all these contributions in the terms of the two functions f(mx(v))

and g(mx(v)), which we define in a moment. The argument mx(v) represents one

of our masses.

The function f(mx(v)) is equal to the one loop zero-temperature potential for

the scalar model, to be specific the function f(mx(v)) is solution of the integral

5.3.

f(mx(v)) =Λ2

32π2m2x(v) +

m4x(v)

64π2

[ln(m2

x(v)

Λ2

)− 1

2

]. (5.16)

The function g(mx(v)) is equal to the one loop finite-temperature potential for

the scalar model

g(mx(v)) =T

2π2


√k2+m2(v)/T ). (5.17)

With these definitions we can write the contributions

V(0)

1,H(v) = f(m1(v)) + 3f(m2(v)),

V(0)

1,gb(v) = 3f(mZ(v)) + 6f(mW (v)),

V(T )

1,H (v) = g(m1(v)) + 3g(m2(v)),

V(T )

1,gb(v) = 3g(mZ(v)) + 6g(mW (v)).

(5.18)

The numerical factors in front come as follows: factor of three in the Higgs-boson

contribution comes from the three fields that have the same mass of m2(v), the

factor three in front of the second expression comes from three polarization states

of the Z-boson, and six comes from the two W-boson of which both have three

polarizations.

Let us now take a closer look at the zero-temperature part of the effective

potential. As we have done before, we can renormalize this by introducing coun-

terterms and then work them out by introducing renormalization conditions. Now

we have case where a part of our potential depends of the cutoff scale. Familiar

counterterms are

Lct =A

2v2 +

B

4v4 + C.


To get rid of the cutoff dependence we need to define A and B to be functions of

the cut off Λ. Thus we define A = a(Λ) + δc2 and B = b(Λ) + δλ. We must now

choose that a(Λ) cancels the infinite coefficient of the v2- term and b(Λ) cancel

the infinite coefficient infront of the v4. Then we can impose the renormalization

condition

dV (0)(v)

dv2|v=〈v〉0 , (5.19)

where V (0)(v) = V0(v) + V(0)

1 (v). From condition 5.19 we get that δc2 = δλc2/λ.

We can choose δλ ∼ λ2 ∼ 0, since the corrections to the tree potential will not

be significant in the high temperature region. What we are now left with is the

zero-temperature part of the effective potential

V (0)(v) = v2(

332π2λc

2 − 12c2)

+ v4[

14λ− 1

64π2

(6λ2 + 3g4

16+ 3(g2+g′2)2

32

)]+ 1

64π2

[6m4

W (v) ln(λv2

c2

)+ 3m4

Z(v) ln(λv2

c2

)+m4

1(v) ln(m2

1(v)

2c2

)+ 3m4

2(v) ln(m2

2(v)

2c2

)].

(5.20)

We now move to the finite temperature part of the calculation. We now do not

want to restrict our calculations to the case m(v)/T 1 and therefore we have

couple of complications. The gauge boson part does not have any difficulties. The

Higgs boson part in turn has calculation problems that result from the fact that

in the integral of 5.17 the Higgs-boson masses can be negative and therefore the

square root is imaginary [49]. To get around these problems we need to combine

the ring diagrams with the one-loop contributions. The masses in the one-loop

expression gets replaced by shifted masses, m2i (v) + πi(0), which are positive for

high enough temperature, T . And in that case we can determine what is the high

enough T .

5.2.3 The Ring Diagram Contribution

The ring diagram contributions come from the Higgs-boson and gauge-boson loops.

We now need to work out polarization tensors in the infrared limit for the Higgs-

boson and gauge-boson. We still calculate the polarization to the leading order in

terms of them(v)/T , even in the case thatm(v)/T ∼ 1 since then V (v)ring ∼ 0 [49].

We first work out what is the polarization tensors in the Higgs case. There are

four Higgs fields and therefore there are four of the polarization tensors of which

three are the same. There are three loops that contribute to these polarization

tensors, the loops come from SU(2) field, U(1) field and the Higgs field, shown in

the figure 5.4.


(a) The Higgs loop

(b) the U(1) loop

(c) the SU(2) loop

Figure 5.4: The loops contributing to the Higgs polarization tensors.

As we obtained the polarization tensor in the previous section 5.10, we get now

for the ıth Higgs field the polarization tensor, πı(0),

πı(0) = π(φ)φ (0) + π

(Aaµ)

φ (0) + π(Bµ)φ (0), (5.21)

whereπ

(φ)φ (0) = λ

2T 2,

π(Aaµ)

φ (0) = g2

8T 2,

π(Bµ)φ (0) = g2+g′2

16T 2,

(5.22)

are the leading order polarization tensors for each of the cases. Thus we get the

ring diagram contribution

V φı(v)ring = −1

2T∑n

∫d3~q

(2π)3

∞∑N=1

1

N

[− 1

ωn + ~q2 +mı

πı

]N. (5.23)

Now we get to combine this with the finite part of the one-loop contribution. For

the Higgs part of the effective potential

V(T )φ (v) = V

(T )1,φ (v) + V φı(v)ring (5.24)

= g(m21(v)) + 3g(m2

2(v)) + V φ1(v)ring + 3V φ2(v)ring.

These functions combine quite easily

= g(m21(v) + π1(0)) + 3g(m2

2(v) + π2(0)), (5.25)

where g is the function defined at equation 5.17. The argument of the square root

is now real for all values of k when m2j(v)+πj(0) ≥ 0, and real for all v if πj(0) ≥ c2.

When π1(0) = T 2

16(8λ+ 3g2 + g′2), we get the condition for the temperature T

T 2 ≥ T 21 =

16 c2

8λ+ 3g2 + g′2. (5.26)


This condition is essentially same as requiring the effective mass of the Higgs boson

to be positive. The effective mass gets its value from the quadratic term of the

effective potential. This part of the effective potential is obtained from the tree

potential and expansion of the finite-temperature part [49]

V (v)quadratic = −c2

2v2 +

T

24[m2

1(v) + 3m22(v) + 6m2

W (v) + 3m2Z(v)]. (5.27)

Then the effective mass

m2effect '

∂2

∂v2V (v)quadratic = πj(0)− c2, (5.28)

which is positive when πj(0) ≥ c2, in other words when T ≥ T1. Within these limits

the Higgs contribution for the effective potential and the Higgs boson effective mass

both are real. It is noted here that for any values of the coupling constants, the

phase transition will be of the first order as long as there is some temperature T1

for which m2eff = 0. Increasing the temperature above T1 will produce positive

curvature at the origin, which indicates a local minimum at the origin and a first

order phase transition.

5.2.4 Gauge-boson Contribution from Ring Diagrams

Now we can move on to the gauge boson part of the effective potential. Let’s

start again from the polarization tensor. We are now working in the original basis

where the relevant fields were Aaµ and Bµ. We now write the polarization tensor in

the matrix form, in the infrared limit polarization tensor is denoted πAB(0). This

expression can be obtained [49] by defining projection operators Tµν and Lµν and

expanding the polarization tensor

πABµν (0) = πABT (0)Tµν + πABL (0)Lµν ,

whereT00 = 0, T0i = Ti0 = 0, Tij = δij − kikj

k2

and Lµν = kµkνk2− ηµν − Tµν .

In the infrared limit

T00 = 0, L00 = 0, T ii = −δii −k2

k2= −2, Lii = 0,

and therefore

πABµν (0) = −LµνπAB00 (0). (5.29)

In the limit mW (v)/T 1 and mZ(v)/T 1, πAB00 (0) is diagonal matrix

π00(0) =

π

(2)00 (0) 0 0 0

0 π(2)00 (0) 0 0

0 0 π(2)00 (0) 0

0 0 0 π(1)00 (0),

(5.30)


where 2 and 1 indicate the polarization tensors for SU(2)- and U(1)- bosons. Di-

agrams that now contribute come from the SU(2) field self interactions and from

the loop diagrams where the Higgs boson does the loop, shown in the figures 5.5

and 5.6.

(a)

(b)

(c)

Figure 5.5: These are SU(2) self interaction diagrams and they contribute for

π(2)gb (0)

(a) π

(2)φ (0)

(b) π

(1)φ (0)

Figure 5.6: The Higgs loops contribute to the polarization tensor.

So the terms in the matrix

π(2)00 (0) = π

(2)gb (0) + π

(2)φ (0) and π

(1)00 (0) = π

(1)φ (0), (5.31)

whereπ

(2)gb (0) = 2 g2

3T 2,

π(2)φ (0) = g2

6T 2,

π(1)φ (0) = g′2

6T 2,

are the leading term in the polarization tensor expansion respect to mj(v)/T .

The gauge boson ring diagram contribution is now given in the matrix form

V gb(v)ring = − ı2

∫d4q

(2π)4Tr

∞∑N=1

1

N

[ıπµνAB(0)ıDBC

νλ (q)]N, (5.32)

5.3. SCALAR THEORY WITH THE GAUGE FIELD 55

where

ıDABµν (q) = ı(Tµν + Lµν)

( 1

q2 +M2(v) + ıε

)AB, (5.33)

is the gauge boson propagator in the Landau gauge. Now from the equations 5.29,

5.32 and 5.33, we can put together

V gb(v)ring = − ı2

(−Lµµ)

∫d4q

(2π)4Tr

∞∑N=1

1

N

[( 1

q2 +M2(v) + ıε

)ABπBC00 (0)

]N,

(5.34)

where trace only acts on the indices A and B. At finite temperature this takes

form

V gb(v)ring = −T2

∫d3q

(2π)3Tr

∞∑N=1

1

N

[( 1

ωn + ~q2 +M2(v)

)ABπBC00 (0)

]N, (5.35)

We can now do this integral and we obtain

V gb(v)ring = − T

12πTr(

[M2(v) + π00(0)]3/2 −M3(v)), (5.36)

where M2(v) was defined in the equation 1.2. Here we can notice now that terms

that are proportional to the m3(v) are not canceled. Thus the cancellation of those

kind of terms in not general property of the calculation of the effective potential.

However for the gauge boson case, the squares of the masses are always positive

for all v and therefore the contribution for the effective potential and gauge boson

masses are always real.

The final expression is obtained by combining V (0)(v), V(T )φ (v), V

(T )1,gb(v) and

V gb(v) in the equations 5.20, 5.25, 5.18 and 5.36. The integrals should be cal-

culated numerically to get the best result. The effective potential is real for the

temperature

T ≥ T1 =16 c2

8λ+ 3g2 + g′2.

When g = 0.637 and g′ = 0.344 and the 〈v〉0 = 246 GeV, correct W - and Z-boson

masses are reproduced. In the Standard Model the Higgs self-coupling constant,

λ, is 0.12.

5.3 Scalar Theory with the Gauge Field

To this state we have discussed the simple scalar theory and the Standard Model

without fermionic contribution. We now can discuss as a special case the same

kind of theory as we have in the 3.2, single scalar field and the U(1) gauge field.

We compute the effective potential as done previous section, but consider only

loops with photons and neglect the W and Z loops. In this section we have the

Lagrangian where mass term of the scalar field is present.


We will use same notation as earlier, U(1) gauge field is denoted by Bµ and

the coupling constant is denoted by g′. The scalar field is denoted by φ.

The Lagrangian density is now

L =1

2(Dµφ)2 +

1

2c2φ2 − 1

4λφ4 − 1

4FµνF

µν , (5.37)

where Dµ = ∂µ + ı2g′Bµ. All the calculations for this case are actually already

done.

5.3.1 Calculation of the Effective Potential

As done earlier we will divide the effective potential calculation in the three parts,

zero-temperature part, finite temperature part and the ring diagram contribution.

Tree level potential stays the same as in the case where we had only one scalar

field

V0(v) = −c2

2v2 +

λ

4v4.

Zero temperature part stays the same as well, thus the tree level and zero temper-

ature renormalized potential is

V (0)(v) = −c2

2v2 +

λ

4v4 +

1

64π2m4(v) ln

(m2(v)

2c2

)+

21λc2

64π2v2 − 27λ2

128π2v4 (5.38)

Finite teperature part of the loop corrections stay as they were in the scalar

theory 5.1

V(T )

1 (v) =T

2π2


√k2+m2(v)/T ), (5.39)

For limit m(v)/T 1, this integral can be now expanded in the terms of m(v)/T

as we did before

V(T )

1 (v) = T 4[−π

2

90+m2(v)

24T 2−m

3(v)

12πT 3− m4(v)

32π2T 4ln(m(v)

4πT

)+O(m4(v)/T 4)

]. (5.40)

The loop diagram contributon differs from the scalar theory. The polarization

tensor is now sum of two diagrams, shown in the figure 5.7.

The scalar loop contributon to the polarization tensor is same as it was in the

scalar case and the U(1) loop is the same as in the Standard Model case,

π(0) = π(φ)φ (0) + π

(Bµ)φ (0), (5.41)

whereπ

(φ)φ (0) = λ

4T 2,

π(Bµ)φ (0) = g′2

16T 2.


(a) The Higgs loop

(b) the photon loop

Figure 5.7: The loops contributing to the scalar polarization tensor.

Thus the scalar ring diagram contribution gives

V φring(v) = − T

12π

[(m2(v) + π(0))3/2 −m3(v)

].

The ring diagram that contributes to the gauge boson loop, show in the figure

5.8. Polarization tensor now is πU(1)(0) = g′2

6T 2. When considering the Lagrangian

Figure 5.8: The scalar loop that contributes to the polarization tensor.

density for this case we find that the Lagrangian holds term

Lgauge =1

8g′2BµB

µφ2,

thus we can determine that ring diagram contribution only for the U(1) gauge field

is

VU(1)ring (v) = − T

12π

[(g′24v2 + πU(1)(0)

)3/2

−(g′2

4v2) 3

2]. (5.42)

Thus the effective potential gets the form

V (v) = − c2

2v2 + λ

4v4 + 21λc2

64π2 v2 − 27λ2

128π2v4

+m2(v)24

T 2 − T12π

(m2(v) + g′2

16T 2 + λ

4T 2)3/2

− T12π

[(g′2

4v2 + g′2

6T 2)3/2

−(g′2

4v2) 3

2]

+O(m4(v)).

(5.43)

As previously the terms proportional to the m4(v) ln(m2(v)) are canceled as are

the terms proportional to m3(v). The two last terms of the effective potential are

the contribution of the gauge field loop.


5.3.2 Conclusions

In section 3.3 we considered theory where we had Standard Model at the visible

sector and a scalar field and a gauge field at the hidden sector. In the previous

section 5.3.1 we have calculated the effective potential for the fields at the hidden

sector. We can now study the behavior of the effective potential as a function of

temperature.

As previously we need to be sure that the effective potential stays positive. For

that reason we must require that argument of square root stays positive,

m2(v) + T 2(g′2

16+λ

4

)= 3λv2 − c2 + T 2(

g′2

16+λ

4) > 0,

this is true for every v, when T ≥ T1 = 2c√λ+ 1

4g′2

.

T1 =2c√

λ+ 14g′2' 2c√

λ− cg′2

4λ3/2+O(g′4).

In this case the effective mass is

m2eff =

∂2

∂v2Veff |v=0 = −c2 +

λ

4T 2 − g′3

16√

6πT 2

For example for the Standard Model values of the Higgs the self-interaction

λ = 0.12, mass parameter |c| = 88.8 GeV [54] and g′ = 0.344 would give us

T1 = 459.2 GeV. At that temperature the tree potential minus sign dominates.

The potential has the form of the black line in figure 5.9. When temperature is

further increased over T1 the sign infront of v2 - term flips and potential gets the

form of the blue line in figure 5.9 and gains minimum at the origin.

v

V (v)

Figure 5.9: At black form of the effective potential 5.43 at T = T1 and at blue is

the form of the potential at T = 491.44 GeV.


At high temperature potential has global minimum at the origin as temperature

drops the potential gains new degenate minimum away from the origin. This

indicates first order phase transition [49, 55]. To get more descriptive discussion

the potential should be properly calculated numerically and plotted more precisely.

As we have seen in section 3.3, the symmetry break is transmitted from the

hidden sector to the visible sector explaining the origin of the weak scale. At finite

temperature similarly the first order phase transition in the hidden sector would

mean first order phase transition at the Standard Model sector [55]. We leave

more careful investigation of this possibility for future work.

Chapter 6

Summary and outlook

In this thesis we started from the quantatization of the scalar fields and electromag-

netic field. We have introduce the interactions between the fields in φ4 -theory as

in the scalar electrodynamics. We have conveniently chosen these theories so that

we can build a basis in which we can introduce the Coleman-Weinberg mechanism.

After that we have introduced in depth the Coleman-Weinberg mechanism and

how the effective potential is derived from the loop corrections. When applying

Coleman-Weinberg mechanism to the Stadard model, predicted Higgs boson mass

is too low. We introduced expansion of the Standard Model, in which at the

hidden sector contains a scalar field and a gauge field to which the Coleman-

Weinberg mechanism is applied. We then explained how the symmetry breaking

would be conveyed to the Standard Model through the Higgs portal. We then

introduced some of the phenomenology of the model.

After that we have introduced the path integral formalism of the thermal field

theory. We did not really go far in the thermal theory because we saw it sufficent

to go through the calculations as they appear in the calculation of the effective

potential.

In the last chapter of this thesis we considered three cases where spontaneously

broken symmetry is restored at high temperature. In the last case we considered

the hidden sector of our previous model. The main goal was to show that when

symmetry is restored the process goes through first order phase transiton. That

first order phase transition is then transferred to the Standard Model fulfilling the

Sakharov’s condition for baryogenesis.

To obtain a better understanding of the behavior of the effective potential as a

function of the temperature, the fintite temperature part of the effective potential,

shown in equation 5.40, should be integrated numerically. The effective potential

should be plotted more precisely in different temperatures to show that system

goes through the first order phase transition.

Nevertheless, our results show that extensions of the Standard Model with

a weakly coupled singlet sector provide an interesting model framework. In our

analysis we have explicitly demonstrated how one can address model building

60

61

paradigms like naturality and also phenomenological issues like the order of the

electroweak transition at finite temperature and its consequences for the generation

of the observed baryon asymmetry.

To further address the shortcomings of the Standard Model and as a concrete

further research direction one can imagine adding fields into the hidden sector. One

viable choice would be adding fermion fields. Such a fermion would constitute a

dark matter candidate or its mixing with the Standard Model leptons could explain

the observed neutrino mass patterns. The computational tools we have reviewed

in this thesis could be adapted to study this case as well.

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